11: Multirate Systems
• Multirate Systems
• Building blocks
• Resampling Cascades
• Noble Identities
• Noble Identities Proof
• Upsampled z-transform
• Downsampled z-transform
• Downsampled Spectrum
• Power Spectral Density
• Perfect Reconstruction
• Commutators
• Summary
• MATLAB routines
DSP and Digital Filters (2016-9045)
11: Multirate Systems
Multirate: 11 – 1 / 14
Multirate Systems
11: Multirate Systems
• Multirate Systems
• Building blocks
• Resampling Cascades
• Noble Identities
• Noble Identities Proof
• Upsampled z-transform
• Downsampled z-transform
• Downsampled Spectrum
• Power Spectral Density
• Perfect Reconstruction
• Commutators
• Summary
• MATLAB routines
Multirate systems include more than one sample rate
DSP and Digital Filters (2016-9045)
Multirate: 11 – 2 / 14
Multirate Systems
11: Multirate Systems
• Multirate Systems
• Building blocks
• Resampling Cascades
• Noble Identities
• Noble Identities Proof
• Upsampled z-transform
• Downsampled z-transform
• Downsampled Spectrum
• Power Spectral Density
• Perfect Reconstruction
• Commutators
• Summary
• MATLAB routines
Multirate systems include more than one sample rate
Why bother?:
• May need to change the sample rate
DSP and Digital Filters (2016-9045)
e.g. Audio sample rates include 32, 44.1, 48, 96 kHz
Multirate: 11 – 2 / 14
Multirate Systems
11: Multirate Systems
• Multirate Systems
• Building blocks
• Resampling Cascades
• Noble Identities
• Noble Identities Proof
• Upsampled z-transform
• Downsampled z-transform
• Downsampled Spectrum
• Power Spectral Density
• Perfect Reconstruction
• Commutators
• Summary
• MATLAB routines
Multirate systems include more than one sample rate
Why bother?:
• May need to change the sample rate
e.g. Audio sample rates include 32, 44.1, 48, 96 kHz
• Can relax analog or digital filter requirements
DSP and Digital Filters (2016-9045)
e.g. Audio DAC increases sample rate so that the reconstruction filter
can have a more gradual cutoff
Multirate: 11 – 2 / 14
Multirate Systems
11: Multirate Systems
• Multirate Systems
• Building blocks
• Resampling Cascades
• Noble Identities
• Noble Identities Proof
• Upsampled z-transform
• Downsampled z-transform
• Downsampled Spectrum
• Power Spectral Density
• Perfect Reconstruction
• Commutators
• Summary
• MATLAB routines
Multirate systems include more than one sample rate
Why bother?:
• May need to change the sample rate
e.g. Audio sample rates include 32, 44.1, 48, 96 kHz
• Can relax analog or digital filter requirements
e.g. Audio DAC increases sample rate so that the reconstruction filter
can have a more gradual cutoff
• Reduce computational complexity
fs
where ∆f is width of transition band
FIR filter length ∝ ∆f
Lower fs ⇒ shorter filter + fewer samples ⇒computation ∝ fs2
DSP and Digital Filters (2016-9045)
Multirate: 11 – 2 / 14
Building blocks
11: Multirate Systems
• Multirate Systems
• Building blocks
• Resampling Cascades
• Noble Identities
• Noble Identities Proof
• Upsampled z-transform
• Downsampled z-transform
• Downsampled Spectrum
• Power Spectral Density
• Perfect Reconstruction
• Commutators
• Summary
• MATLAB routines
Downsample
DSP and Digital Filters (2016-9045)
y[m] = x[Km]
Multirate: 11 – 3 / 14
Building blocks
11: Multirate Systems
• Multirate Systems
• Building blocks
• Resampling Cascades
• Noble Identities
• Noble Identities Proof
• Upsampled z-transform
• Downsampled z-transform
• Downsampled Spectrum
• Power Spectral Density
• Perfect Reconstruction
• Commutators
• Summary
• MATLAB routines
Downsample
Upsample
DSP and Digital Filters (2016-9045)
y[m] = x[Km]
( n
u K
K|n
v[n] =
0
else
Multirate: 11 – 3 / 14
Building blocks
11: Multirate Systems
• Multirate Systems
• Building blocks
• Resampling Cascades
• Noble Identities
• Noble Identities Proof
• Upsampled z-transform
• Downsampled z-transform
• Downsampled Spectrum
• Power Spectral Density
• Perfect Reconstruction
• Commutators
• Summary
• MATLAB routines
y[m] = x[Km]
( n
u K
K|n
v[n] =
0
else
Downsample
Upsample
Example:
Downsample by 3 then upsample by 4
w[n]
0
DSP and Digital Filters (2016-9045)
Multirate: 11 – 3 / 14
Building blocks
11: Multirate Systems
• Multirate Systems
• Building blocks
• Resampling Cascades
• Noble Identities
• Noble Identities Proof
• Upsampled z-transform
• Downsampled z-transform
• Downsampled Spectrum
• Power Spectral Density
• Perfect Reconstruction
• Commutators
• Summary
• MATLAB routines
y[m] = x[Km]
( n
u K
K|n
v[n] =
0
else
Downsample
Upsample
Example:
Downsample by 3 then upsample by 4
w[n]
0
DSP and Digital Filters (2016-9045)
x[m]
0
Multirate: 11 – 3 / 14
Building blocks
11: Multirate Systems
• Multirate Systems
• Building blocks
• Resampling Cascades
• Noble Identities
• Noble Identities Proof
• Upsampled z-transform
• Downsampled z-transform
• Downsampled Spectrum
• Power Spectral Density
• Perfect Reconstruction
• Commutators
• Summary
• MATLAB routines
y[m] = x[Km]
( n
u K
K|n
v[n] =
0
else
Downsample
Upsample
Example:
Downsample by 3 then upsample by 4
w[n]
0
DSP and Digital Filters (2016-9045)
x[m]
0
y[r]
0
Multirate: 11 – 3 / 14
Building blocks
11: Multirate Systems
• Multirate Systems
• Building blocks
• Resampling Cascades
• Noble Identities
• Noble Identities Proof
• Upsampled z-transform
• Downsampled z-transform
• Downsampled Spectrum
• Power Spectral Density
• Perfect Reconstruction
• Commutators
• Summary
• MATLAB routines
y[m] = x[Km]
( n
u K
K|n
v[n] =
0
else
Downsample
Upsample
Example:
Downsample by 3 then upsample by 4
w[n]
0
x[m]
0
y[r]
0
• We use different index variables (n, m, r ) for different sample rates
DSP and Digital Filters (2016-9045)
Multirate: 11 – 3 / 14
Building blocks
11: Multirate Systems
• Multirate Systems
• Building blocks
• Resampling Cascades
• Noble Identities
• Noble Identities Proof
• Upsampled z-transform
• Downsampled z-transform
• Downsampled Spectrum
• Power Spectral Density
• Perfect Reconstruction
• Commutators
• Summary
• MATLAB routines
y[m] = x[Km]
( n
u K
K|n
v[n] =
0
else
Downsample
Upsample
Example:
Downsample by 3 then upsample by 4
w[n]
0
x[m]
0
y[r]
0
• We use different index variables (n, m, r ) for different sample rates
• Use different colours for signals at different rates (sometimes)
DSP and Digital Filters (2016-9045)
Multirate: 11 – 3 / 14
Building blocks
11: Multirate Systems
• Multirate Systems
• Building blocks
• Resampling Cascades
• Noble Identities
• Noble Identities Proof
• Upsampled z-transform
• Downsampled z-transform
• Downsampled Spectrum
• Power Spectral Density
• Perfect Reconstruction
• Commutators
• Summary
• MATLAB routines
y[m] = x[Km]
( n
u K
K|n
v[n] =
0
else
Downsample
Upsample
Example:
Downsample by 3 then upsample by 4
w[n]
0
x[m]
0
y[r]
0
• We use different index variables (n, m, r ) for different sample rates
• Use different colours for signals at different rates (sometimes)
• Synchronization: all signals have a sample at n = 0.
DSP and Digital Filters (2016-9045)
Multirate: 11 – 3 / 14
Resampling Cascades
11: Multirate Systems
• Multirate Systems
• Building blocks
• Resampling Cascades
• Noble Identities
• Noble Identities Proof
• Upsampled z-transform
• Downsampled z-transform
• Downsampled Spectrum
• Power Spectral Density
• Perfect Reconstruction
• Commutators
• Summary
• MATLAB routines
Successive downsamplers or upsamplers
can be combined
DSP and Digital Filters (2016-9045)
Multirate: 11 – 4 / 14
Resampling Cascades
11: Multirate Systems
• Multirate Systems
• Building blocks
• Resampling Cascades
• Noble Identities
• Noble Identities Proof
• Upsampled z-transform
• Downsampled z-transform
• Downsampled Spectrum
• Power Spectral Density
• Perfect Reconstruction
• Commutators
• Summary
• MATLAB routines
Successive downsamplers or upsamplers
can be combined
Upsampling can be exactly inverted
DSP and Digital Filters (2016-9045)
Multirate: 11 – 4 / 14
Resampling Cascades
11: Multirate Systems
• Multirate Systems
• Building blocks
• Resampling Cascades
• Noble Identities
• Noble Identities Proof
• Upsampled z-transform
• Downsampled z-transform
• Downsampled Spectrum
• Power Spectral Density
• Perfect Reconstruction
• Commutators
• Summary
• MATLAB routines
Successive downsamplers or upsamplers
can be combined
Upsampling can be exactly inverted
Downsampling destroys information
permanently ⇒ uninvertible
DSP and Digital Filters (2016-9045)
Multirate: 11 – 4 / 14
Resampling Cascades
11: Multirate Systems
• Multirate Systems
• Building blocks
• Resampling Cascades
• Noble Identities
• Noble Identities Proof
• Upsampled z-transform
• Downsampled z-transform
• Downsampled Spectrum
• Power Spectral Density
• Perfect Reconstruction
• Commutators
• Summary
• MATLAB routines
Successive downsamplers or upsamplers
can be combined
Upsampling can be exactly inverted
Downsampling destroys information
permanently ⇒ uninvertible
Resampling can be interchanged
iff P and Q are coprime (surprising!)
DSP and Digital Filters (2016-9045)
Multirate: 11 – 4 / 14
Resampling Cascades
11: Multirate Systems
• Multirate Systems
• Building blocks
• Resampling Cascades
• Noble Identities
• Noble Identities Proof
• Upsampled z-transform
• Downsampled z-transform
• Downsampled Spectrum
• Power Spectral Density
• Perfect Reconstruction
• Commutators
• Summary
• MATLAB routines
Successive downsamplers or upsamplers
can be combined
Upsampling can be exactly inverted
Downsampling destroys information
permanently ⇒ uninvertible
Resampling can be interchanged
iff P and Q are coprime (surprising!)
Proof: Left side: y[n] = w
h
1
Qn
i
=x
h
P
Qn
i
if Q | n else y[n] = 0.
[Note: a | b means “a divides into b exactly”]
DSP and Digital Filters (2016-9045)
Multirate: 11 – 4 / 14
Resampling Cascades
11: Multirate Systems
• Multirate Systems
• Building blocks
• Resampling Cascades
• Noble Identities
• Noble Identities Proof
• Upsampled z-transform
• Downsampled z-transform
• Downsampled Spectrum
• Power Spectral Density
• Perfect Reconstruction
• Commutators
• Summary
• MATLAB routines
Successive downsamplers or upsamplers
can be combined
Upsampling can be exactly inverted
Downsampling destroys information
permanently ⇒ uninvertible
Resampling can be interchanged
iff P and Q are coprime (surprising!)
Proof: Left side: y[n] = w
h
1
Qn
i
=x
h
h
P
Qn
i
if Q | n else y[n] = 0.
i
P
Right side: v[n] = u [P n] = x Q
n if Q | P n.
[Note: a | b means “a divides into b exactly”]
DSP and Digital Filters (2016-9045)
Multirate: 11 – 4 / 14
Resampling Cascades
11: Multirate Systems
• Multirate Systems
• Building blocks
• Resampling Cascades
• Noble Identities
• Noble Identities Proof
• Upsampled z-transform
• Downsampled z-transform
• Downsampled Spectrum
• Power Spectral Density
• Perfect Reconstruction
• Commutators
• Summary
• MATLAB routines
Successive downsamplers or upsamplers
can be combined
Upsampling can be exactly inverted
Downsampling destroys information
permanently ⇒ uninvertible
Resampling can be interchanged
iff P and Q are coprime (surprising!)
Proof: Left side: y[n] = w
h
1
Qn
i
=x
h
h
P
Qn
i
if Q | n else y[n] = 0.
i
P
Right side: v[n] = u [P n] = x Q
n if Q | P n.
But {Q | P n ⇒ Q | n} iff P and Q are coprime.
[Note: a | b means “a divides into b exactly”]
DSP and Digital Filters (2016-9045)
Multirate: 11 – 4 / 14
Noble Identities
11: Multirate Systems
• Multirate Systems
• Building blocks
• Resampling Cascades
• Noble Identities
• Noble Identities Proof
• Upsampled z-transform
• Downsampled z-transform
• Downsampled Spectrum
• Power Spectral Density
• Perfect Reconstruction
• Commutators
• Summary
• MATLAB routines
Resamplers commute with addition
and multiplication
DSP and Digital Filters (2016-9045)
Multirate: 11 – 5 / 14
Noble Identities
11: Multirate Systems
• Multirate Systems
• Building blocks
• Resampling Cascades
• Noble Identities
• Noble Identities Proof
• Upsampled z-transform
• Downsampled z-transform
• Downsampled Spectrum
• Power Spectral Density
• Perfect Reconstruction
• Commutators
• Summary
• MATLAB routines
Resamplers commute with addition
and multiplication
Delays must be multiplied by the
resampling ratio
DSP and Digital Filters (2016-9045)
Multirate: 11 – 5 / 14
Noble Identities
11: Multirate Systems
• Multirate Systems
• Building blocks
• Resampling Cascades
• Noble Identities
• Noble Identities Proof
• Upsampled z-transform
• Downsampled z-transform
• Downsampled Spectrum
• Power Spectral Density
• Perfect Reconstruction
• Commutators
• Summary
• MATLAB routines
Resamplers commute with addition
and multiplication
Delays must be multiplied by the
resampling ratio
Noble identities:
Exchange resamplers and filters
DSP and Digital Filters (2016-9045)
Multirate: 11 – 5 / 14
Noble Identities
11: Multirate Systems
• Multirate Systems
• Building blocks
• Resampling Cascades
• Noble Identities
• Noble Identities Proof
• Upsampled z-transform
• Downsampled z-transform
• Downsampled Spectrum
• Power Spectral Density
• Perfect Reconstruction
• Commutators
• Summary
• MATLAB routines
Resamplers commute with addition
and multiplication
Delays must be multiplied by the
resampling ratio
Noble identities:
Exchange resamplers and filters
Example: H(z) = h[0] + h[1]z −1 + h[2]z −2 + · · ·
H(z 3 ) = h[0] + h[1]z −3 + h[2]z −6 + · · ·
DSP and Digital Filters (2016-9045)
Multirate: 11 – 5 / 14
Noble Identities
11: Multirate Systems
• Multirate Systems
• Building blocks
• Resampling Cascades
• Noble Identities
• Noble Identities Proof
• Upsampled z-transform
• Downsampled z-transform
• Downsampled Spectrum
• Power Spectral Density
• Perfect Reconstruction
• Commutators
• Summary
• MATLAB routines
Resamplers commute with addition
and multiplication
Delays must be multiplied by the
resampling ratio
Noble identities:
Exchange resamplers and filters
Corrollary
Example: H(z) = h[0] + h[1]z −1 + h[2]z −2 + · · ·
H(z 3 ) = h[0] + h[1]z −3 + h[2]z −6 + · · ·
DSP and Digital Filters (2016-9045)
Multirate: 11 – 5 / 14
Noble Identities Proof
11: Multirate Systems
• Multirate Systems
• Building blocks
• Resampling Cascades
• Noble Identities
• Noble Identities Proof
• Upsampled z-transform
• Downsampled z-transform
• Downsampled Spectrum
• Power Spectral Density
• Perfect Reconstruction
• Commutators
• Summary
• MATLAB routines
Define hQ [n] to be the
impulse response of H(z Q ).
DSP and Digital Filters (2016-9045)
Multirate: 11 – 6 / 14
Noble Identities Proof
11: Multirate Systems
• Multirate Systems
• Building blocks
• Resampling Cascades
• Noble Identities
• Noble Identities Proof
• Upsampled z-transform
• Downsampled z-transform
• Downsampled Spectrum
• Power Spectral Density
• Perfect Reconstruction
• Commutators
• Summary
• MATLAB routines
Define hQ [n] to be the
impulse response of H(z Q ).
Assume that h[r] is of length M + 1 so that hQ [n] is of length QM + 1.
DSP and Digital Filters (2016-9045)
Multirate: 11 – 6 / 14
Noble Identities Proof
11: Multirate Systems
• Multirate Systems
• Building blocks
• Resampling Cascades
• Noble Identities
• Noble Identities Proof
• Upsampled z-transform
• Downsampled z-transform
• Downsampled Spectrum
• Power Spectral Density
• Perfect Reconstruction
• Commutators
• Summary
• MATLAB routines
Define hQ [n] to be the
impulse response of H(z Q ).
Assume that h[r] is of length M + 1 so that hQ [n] is of length QM + 1.
We know that hQ [n] = 0 except when Q | n and that h[r] = hQ [Qr].
DSP and Digital Filters (2016-9045)
Multirate: 11 – 6 / 14
Noble Identities Proof
11: Multirate Systems
• Multirate Systems
• Building blocks
• Resampling Cascades
• Noble Identities
• Noble Identities Proof
• Upsampled z-transform
• Downsampled z-transform
• Downsampled Spectrum
• Power Spectral Density
• Perfect Reconstruction
• Commutators
• Summary
• MATLAB routines
Define hQ [n] to be the
impulse response of H(z Q ).
Assume that h[r] is of length M + 1 so that hQ [n] is of length QM + 1.
We know that hQ [n] = 0 except when Q | n and that h[r] = hQ [Qr].
w[r] = v[Qr]
DSP and Digital Filters (2016-9045)
Multirate: 11 – 6 / 14
Noble Identities Proof
11: Multirate Systems
• Multirate Systems
• Building blocks
• Resampling Cascades
• Noble Identities
• Noble Identities Proof
• Upsampled z-transform
• Downsampled z-transform
• Downsampled Spectrum
• Power Spectral Density
• Perfect Reconstruction
• Commutators
• Summary
• MATLAB routines
Define hQ [n] to be the
impulse response of H(z Q ).
Assume that h[r] is of length M + 1 so that hQ [n] is of length QM + 1.
We know that hQ [n] = 0 except when Q | n and that h[r] = hQ [Qr].
w[r] = v[Qr] =
DSP and Digital Filters (2016-9045)
PQM
s=0
hQ [s]x[Qr − s]
Multirate: 11 – 6 / 14
Noble Identities Proof
11: Multirate Systems
• Multirate Systems
• Building blocks
• Resampling Cascades
• Noble Identities
• Noble Identities Proof
• Upsampled z-transform
• Downsampled z-transform
• Downsampled Spectrum
• Power Spectral Density
• Perfect Reconstruction
• Commutators
• Summary
• MATLAB routines
Define hQ [n] to be the
impulse response of H(z Q ).
Assume that h[r] is of length M + 1 so that hQ [n] is of length QM + 1.
We know that hQ [n] = 0 except when Q | n and that h[r] = hQ [Qr].
PQM
w[r] = v[Qr] = s=0 hQ [s]x[Qr − s]
PM
= m=0 hQ [Qm]x[Qr − Qm]
DSP and Digital Filters (2016-9045)
Multirate: 11 – 6 / 14
Noble Identities Proof
11: Multirate Systems
• Multirate Systems
• Building blocks
• Resampling Cascades
• Noble Identities
• Noble Identities Proof
• Upsampled z-transform
• Downsampled z-transform
• Downsampled Spectrum
• Power Spectral Density
• Perfect Reconstruction
• Commutators
• Summary
• MATLAB routines
Define hQ [n] to be the
impulse response of H(z Q ).
Assume that h[r] is of length M + 1 so that hQ [n] is of length QM + 1.
We know that hQ [n] = 0 except when Q | n and that h[r] = hQ [Qr].
PQM
w[r] = v[Qr] = s=0 hQ [s]x[Qr − s]
PM
PM
= m=0 hQ [Qm]x[Qr − Qm] = m=0 h[m]x[Q(r − m)]
DSP and Digital Filters (2016-9045)
Multirate: 11 – 6 / 14
Noble Identities Proof
11: Multirate Systems
• Multirate Systems
• Building blocks
• Resampling Cascades
• Noble Identities
• Noble Identities Proof
• Upsampled z-transform
• Downsampled z-transform
• Downsampled Spectrum
• Power Spectral Density
• Perfect Reconstruction
• Commutators
• Summary
• MATLAB routines
Define hQ [n] to be the
impulse response of H(z Q ).
Assume that h[r] is of length M + 1 so that hQ [n] is of length QM + 1.
We know that hQ [n] = 0 except when Q | n and that h[r] = hQ [Qr].
PQM
w[r] = v[Qr] = s=0 hQ [s]x[Qr − s]
PM
PM
= m=0 hQ [Qm]x[Qr − Qm] = m=0 h[m]x[Q(r − m)]
PM
= m=0 h[m]u[r − m]
DSP and Digital Filters (2016-9045)
Multirate: 11 – 6 / 14
Noble Identities Proof
11: Multirate Systems
• Multirate Systems
• Building blocks
• Resampling Cascades
• Noble Identities
• Noble Identities Proof
• Upsampled z-transform
• Downsampled z-transform
• Downsampled Spectrum
• Power Spectral Density
• Perfect Reconstruction
• Commutators
• Summary
• MATLAB routines
Define hQ [n] to be the
impulse response of H(z Q ).
Assume that h[r] is of length M + 1 so that hQ [n] is of length QM + 1.
We know that hQ [n] = 0 except when Q | n and that h[r] = hQ [Qr].
PQM
w[r] = v[Qr] = s=0 hQ [s]x[Qr − s]
PM
PM
= m=0 hQ [Qm]x[Qr − Qm] = m=0 h[m]x[Q(r − m)]
PM
,
= m=0 h[m]u[r − m] = y[r]
DSP and Digital Filters (2016-9045)
Multirate: 11 – 6 / 14
Noble Identities Proof
11: Multirate Systems
• Multirate Systems
• Building blocks
• Resampling Cascades
• Noble Identities
• Noble Identities Proof
• Upsampled z-transform
• Downsampled z-transform
• Downsampled Spectrum
• Power Spectral Density
• Perfect Reconstruction
• Commutators
• Summary
• MATLAB routines
Define hQ [n] to be the
impulse response of H(z Q ).
Assume that h[r] is of length M + 1 so that hQ [n] is of length QM + 1.
We know that hQ [n] = 0 except when Q | n and that h[r] = hQ [Qr].
PQM
w[r] = v[Qr] = s=0 hQ [s]x[Qr − s]
PM
PM
= m=0 hQ [Qm]x[Qr − Qm] = m=0 h[m]x[Q(r − m)]
PM
,
= m=0 h[m]u[r − m] = y[r]
Upsampled Noble Identity:
DSP and Digital Filters (2016-9045)
Multirate: 11 – 6 / 14
Noble Identities Proof
11: Multirate Systems
• Multirate Systems
• Building blocks
• Resampling Cascades
• Noble Identities
• Noble Identities Proof
• Upsampled z-transform
• Downsampled z-transform
• Downsampled Spectrum
• Power Spectral Density
• Perfect Reconstruction
• Commutators
• Summary
• MATLAB routines
Define hQ [n] to be the
impulse response of H(z Q ).
Assume that h[r] is of length M + 1 so that hQ [n] is of length QM + 1.
We know that hQ [n] = 0 except when Q | n and that h[r] = hQ [Qr].
PQM
w[r] = v[Qr] = s=0 hQ [s]x[Qr − s]
PM
PM
= m=0 hQ [Qm]x[Qr − Qm] = m=0 h[m]x[Q(r − m)]
PM
,
= m=0 h[m]u[r − m] = y[r]
Upsampled Noble Identity:
We know that v[n] = 0 except when Q | n and that v[Qr] = x[r].
DSP and Digital Filters (2016-9045)
Multirate: 11 – 6 / 14
Noble Identities Proof
11: Multirate Systems
• Multirate Systems
• Building blocks
• Resampling Cascades
• Noble Identities
• Noble Identities Proof
• Upsampled z-transform
• Downsampled z-transform
• Downsampled Spectrum
• Power Spectral Density
• Perfect Reconstruction
• Commutators
• Summary
• MATLAB routines
Define hQ [n] to be the
impulse response of H(z Q ).
Assume that h[r] is of length M + 1 so that hQ [n] is of length QM + 1.
We know that hQ [n] = 0 except when Q | n and that h[r] = hQ [Qr].
PQM
w[r] = v[Qr] = s=0 hQ [s]x[Qr − s]
PM
PM
= m=0 hQ [Qm]x[Qr − Qm] = m=0 h[m]x[Q(r − m)]
PM
,
= m=0 h[m]u[r − m] = y[r]
Upsampled Noble Identity:
We know that v[n] = 0 except when Q | n and that v[Qr] = x[r].
w[n] =
DSP and Digital Filters (2016-9045)
PQM
s=0
hQ [s]v[n − s]
Multirate: 11 – 6 / 14
Noble Identities Proof
11: Multirate Systems
• Multirate Systems
• Building blocks
• Resampling Cascades
• Noble Identities
• Noble Identities Proof
• Upsampled z-transform
• Downsampled z-transform
• Downsampled Spectrum
• Power Spectral Density
• Perfect Reconstruction
• Commutators
• Summary
• MATLAB routines
Define hQ [n] to be the
impulse response of H(z Q ).
Assume that h[r] is of length M + 1 so that hQ [n] is of length QM + 1.
We know that hQ [n] = 0 except when Q | n and that h[r] = hQ [Qr].
PQM
w[r] = v[Qr] = s=0 hQ [s]x[Qr − s]
PM
PM
= m=0 hQ [Qm]x[Qr − Qm] = m=0 h[m]x[Q(r − m)]
PM
,
= m=0 h[m]u[r − m] = y[r]
Upsampled Noble Identity:
We know that v[n] = 0 except when Q | n and that v[Qr] = x[r].
w[n] =
DSP and Digital Filters (2016-9045)
PQM
s=0 hQ [s]v[n − s] =
PM
m=0 hQ [Qm]v[n
− Qm]
Multirate: 11 – 6 / 14
Noble Identities Proof
11: Multirate Systems
• Multirate Systems
• Building blocks
• Resampling Cascades
• Noble Identities
• Noble Identities Proof
• Upsampled z-transform
• Downsampled z-transform
• Downsampled Spectrum
• Power Spectral Density
• Perfect Reconstruction
• Commutators
• Summary
• MATLAB routines
Define hQ [n] to be the
impulse response of H(z Q ).
Assume that h[r] is of length M + 1 so that hQ [n] is of length QM + 1.
We know that hQ [n] = 0 except when Q | n and that h[r] = hQ [Qr].
PQM
w[r] = v[Qr] = s=0 hQ [s]x[Qr − s]
PM
PM
= m=0 hQ [Qm]x[Qr − Qm] = m=0 h[m]x[Q(r − m)]
PM
,
= m=0 h[m]u[r − m] = y[r]
Upsampled Noble Identity:
We know that v[n] = 0 except when Q | n and that v[Qr] = x[r].
PQM
PM
w[n] = s=0 hQ [s]v[n − s] = m=0 hQ [Qm]v[n − Qm]
PM
= m=0 h[m]v[n − Qm]
DSP and Digital Filters (2016-9045)
Multirate: 11 – 6 / 14
Noble Identities Proof
11: Multirate Systems
• Multirate Systems
• Building blocks
• Resampling Cascades
• Noble Identities
• Noble Identities Proof
• Upsampled z-transform
• Downsampled z-transform
• Downsampled Spectrum
• Power Spectral Density
• Perfect Reconstruction
• Commutators
• Summary
• MATLAB routines
Define hQ [n] to be the
impulse response of H(z Q ).
Assume that h[r] is of length M + 1 so that hQ [n] is of length QM + 1.
We know that hQ [n] = 0 except when Q | n and that h[r] = hQ [Qr].
PQM
w[r] = v[Qr] = s=0 hQ [s]x[Qr − s]
PM
PM
= m=0 hQ [Qm]x[Qr − Qm] = m=0 h[m]x[Q(r − m)]
PM
,
= m=0 h[m]u[r − m] = y[r]
Upsampled Noble Identity:
We know that v[n] = 0 except when Q | n and that v[Qr] = x[r].
PQM
PM
w[n] = s=0 hQ [s]v[n − s] = m=0 hQ [Qm]v[n − Qm]
PM
= m=0 h[m]v[n − Qm]
If Q ∤ n, then v[n − Qm] = 0 ∀m so w[n] = 0 = y[n]
DSP and Digital Filters (2016-9045)
Multirate: 11 – 6 / 14
Noble Identities Proof
11: Multirate Systems
• Multirate Systems
• Building blocks
• Resampling Cascades
• Noble Identities
• Noble Identities Proof
• Upsampled z-transform
• Downsampled z-transform
• Downsampled Spectrum
• Power Spectral Density
• Perfect Reconstruction
• Commutators
• Summary
• MATLAB routines
Define hQ [n] to be the
impulse response of H(z Q ).
Assume that h[r] is of length M + 1 so that hQ [n] is of length QM + 1.
We know that hQ [n] = 0 except when Q | n and that h[r] = hQ [Qr].
PQM
w[r] = v[Qr] = s=0 hQ [s]x[Qr − s]
PM
PM
= m=0 hQ [Qm]x[Qr − Qm] = m=0 h[m]x[Q(r − m)]
PM
,
= m=0 h[m]u[r − m] = y[r]
Upsampled Noble Identity:
We know that v[n] = 0 except when Q | n and that v[Qr] = x[r].
PQM
PM
w[n] = s=0 hQ [s]v[n − s] = m=0 hQ [Qm]v[n − Qm]
PM
= m=0 h[m]v[n − Qm]
If Q ∤ n, then v[n − Qm] = 0 ∀m so w[n] = 0 = y[n]
If Q | n = Qr , then w[Qr] =
DSP and Digital Filters (2016-9045)
PM
m=0
h[m]v[Qr − Qm]
Multirate: 11 – 6 / 14
Noble Identities Proof
11: Multirate Systems
• Multirate Systems
• Building blocks
• Resampling Cascades
• Noble Identities
• Noble Identities Proof
• Upsampled z-transform
• Downsampled z-transform
• Downsampled Spectrum
• Power Spectral Density
• Perfect Reconstruction
• Commutators
• Summary
• MATLAB routines
Define hQ [n] to be the
impulse response of H(z Q ).
Assume that h[r] is of length M + 1 so that hQ [n] is of length QM + 1.
We know that hQ [n] = 0 except when Q | n and that h[r] = hQ [Qr].
PQM
w[r] = v[Qr] = s=0 hQ [s]x[Qr − s]
PM
PM
= m=0 hQ [Qm]x[Qr − Qm] = m=0 h[m]x[Q(r − m)]
PM
,
= m=0 h[m]u[r − m] = y[r]
Upsampled Noble Identity:
We know that v[n] = 0 except when Q | n and that v[Qr] = x[r].
PQM
PM
w[n] = s=0 hQ [s]v[n − s] = m=0 hQ [Qm]v[n − Qm]
PM
= m=0 h[m]v[n − Qm]
If Q ∤ n, then v[n − Qm] = 0 ∀m so w[n] = 0 = y[n]
If Q | n = Qr , then w[Qr] =
=
DSP and Digital Filters (2016-9045)
PM
h[m]v[Qr − Qm]
m=0 h[m]x[r − m] = u[r]
PM
m=0
Multirate: 11 – 6 / 14
Noble Identities Proof
11: Multirate Systems
• Multirate Systems
• Building blocks
• Resampling Cascades
• Noble Identities
• Noble Identities Proof
• Upsampled z-transform
• Downsampled z-transform
• Downsampled Spectrum
• Power Spectral Density
• Perfect Reconstruction
• Commutators
• Summary
• MATLAB routines
Define hQ [n] to be the
impulse response of H(z Q ).
Assume that h[r] is of length M + 1 so that hQ [n] is of length QM + 1.
We know that hQ [n] = 0 except when Q | n and that h[r] = hQ [Qr].
PQM
w[r] = v[Qr] = s=0 hQ [s]x[Qr − s]
PM
PM
= m=0 hQ [Qm]x[Qr − Qm] = m=0 h[m]x[Q(r − m)]
PM
,
= m=0 h[m]u[r − m] = y[r]
Upsampled Noble Identity:
We know that v[n] = 0 except when Q | n and that v[Qr] = x[r].
PQM
PM
w[n] = s=0 hQ [s]v[n − s] = m=0 hQ [Qm]v[n − Qm]
PM
= m=0 h[m]v[n − Qm]
If Q ∤ n, then v[n − Qm] = 0 ∀m so w[n] = 0 = y[n]
If Q | n = Qr , then w[Qr] =
=
DSP and Digital Filters (2016-9045)
PM
h[m]v[Qr − Qm]
m=0 h[m]x[r − m] = u[r] = y[Qr]
PM
m=0
,
Multirate: 11 – 6 / 14
Upsampled z-transform
11: Multirate Systems
• Multirate Systems
• Building blocks
• Resampling Cascades
• Noble Identities
• Noble Identities Proof
• Upsampled z-transform
• Downsampled z-transform
• Downsampled Spectrum
• Power Spectral Density
• Perfect Reconstruction
• Commutators
• Summary
• MATLAB routines
V (z) =
DSP and Digital Filters (2016-9045)
−n
v[n]z
n
P
Multirate: 11 – 7 / 14
Upsampled z-transform
11: Multirate Systems
• Multirate Systems
• Building blocks
• Resampling Cascades
• Noble Identities
• Noble Identities Proof
• Upsampled z-transform
• Downsampled z-transform
• Downsampled Spectrum
• Power Spectral Density
• Perfect Reconstruction
• Commutators
• Summary
• MATLAB routines
V (z) =
DSP and Digital Filters (2016-9045)
P
n v[n]z
−n
=
P
n −n
u[
n s.t. K|n
K ]z
Multirate: 11 – 7 / 14
Upsampled z-transform
11: Multirate Systems
• Multirate Systems
• Building blocks
• Resampling Cascades
• Noble Identities
• Noble Identities Proof
• Upsampled z-transform
• Downsampled z-transform
• Downsampled Spectrum
• Power Spectral Density
• Perfect Reconstruction
• Commutators
• Summary
• MATLAB routines
V (z) =
DSP and Digital Filters (2016-9045)
P
−n
P
=
n v[n]z
P
= m u[m]z −Km
n −n
u[
n s.t. K|n
K ]z
Multirate: 11 – 7 / 14
Upsampled z-transform
11: Multirate Systems
• Multirate Systems
• Building blocks
• Resampling Cascades
• Noble Identities
• Noble Identities Proof
• Upsampled z-transform
• Downsampled z-transform
• Downsampled Spectrum
• Power Spectral Density
• Perfect Reconstruction
• Commutators
• Summary
• MATLAB routines
DSP and Digital Filters (2016-9045)
−n
n −n
v[n]z
=
u[
n
n s.t. K|n
K ]z
P
= m u[m]z −Km = U (z K )
V (z) =
P
P
Multirate: 11 – 7 / 14
Upsampled z-transform
11: Multirate Systems
• Multirate Systems
• Building blocks
• Resampling Cascades
• Noble Identities
• Noble Identities Proof
• Upsampled z-transform
• Downsampled z-transform
• Downsampled Spectrum
• Power Spectral Density
• Perfect Reconstruction
• Commutators
• Summary
• MATLAB routines
−n
n −n
v[n]z
=
u[
n
n s.t. K|n
K ]z
P
= m u[m]z −Km = U (z K )
V (z) =
P
P
Spectrum: V (ejω ) = U (ejKω )
DSP and Digital Filters (2016-9045)
Multirate: 11 – 7 / 14
Upsampled z-transform
11: Multirate Systems
• Multirate Systems
• Building blocks
• Resampling Cascades
• Noble Identities
• Noble Identities Proof
• Upsampled z-transform
• Downsampled z-transform
• Downsampled Spectrum
• Power Spectral Density
• Perfect Reconstruction
• Commutators
• Summary
• MATLAB routines
−n
n −n
v[n]z
=
u[
n
n s.t. K|n
K ]z
P
= m u[m]z −Km = U (z K )
V (z) =
P
P
Spectrum: V (ejω ) = U (ejKω )
Spectrum is horizontally shrunk and replicated K times.
DSP and Digital Filters (2016-9045)
Multirate: 11 – 7 / 14
Upsampled z-transform
11: Multirate Systems
• Multirate Systems
• Building blocks
• Resampling Cascades
• Noble Identities
• Noble Identities Proof
• Upsampled z-transform
• Downsampled z-transform
• Downsampled Spectrum
• Power Spectral Density
• Perfect Reconstruction
• Commutators
• Summary
• MATLAB routines
−n
n −n
v[n]z
=
u[
n
n s.t. K|n
K ]z
P
= m u[m]z −Km = U (z K )
V (z) =
P
P
Spectrum: V (ejω ) = U (ejKω )
Spectrum is horizontally shrunk and replicated K times.
Example:
1
0.5
0
DSP and Digital Filters (2016-9045)
-2
0
ω
2
Multirate: 11 – 7 / 14
Upsampled z-transform
11: Multirate Systems
• Multirate Systems
• Building blocks
• Resampling Cascades
• Noble Identities
• Noble Identities Proof
• Upsampled z-transform
• Downsampled z-transform
• Downsampled Spectrum
• Power Spectral Density
• Perfect Reconstruction
• Commutators
• Summary
• MATLAB routines
−n
n −n
v[n]z
=
u[
n
n s.t. K|n
K ]z
P
= m u[m]z −Km = U (z K )
V (z) =
P
P
Spectrum: V (ejω ) = U (ejKω )
Spectrum is horizontally shrunk and replicated K times.
Example:
K = 3: three images of the original spectrum in all.
1
1
0.5
0.5
0
DSP and Digital Filters (2016-9045)
-2
0
ω
2
0
-2
0
ω
2
Multirate: 11 – 7 / 14
Upsampled z-transform
11: Multirate Systems
• Multirate Systems
• Building blocks
• Resampling Cascades
• Noble Identities
• Noble Identities Proof
• Upsampled z-transform
• Downsampled z-transform
• Downsampled Spectrum
• Power Spectral Density
• Perfect Reconstruction
• Commutators
• Summary
• MATLAB routines
−n
n −n
v[n]z
=
u[
n
n s.t. K|n
K ]z
P
= m u[m]z −Km = U (z K )
V (z) =
P
P
Spectrum: V (ejω ) = U (ejKω )
Spectrum is horizontally shrunk and replicated K times.
1
Total energy unchanged; power (= energy/sample) multiplied by K
Example:
K = 3: three images of the original spectrum in all.
1
1
0.5
0.5
0
DSP and Digital Filters (2016-9045)
-2
0
ω
2
0
-2
0
ω
2
Multirate: 11 – 7 / 14
Upsampled z-transform
11: Multirate Systems
• Multirate Systems
• Building blocks
• Resampling Cascades
• Noble Identities
• Noble Identities Proof
• Upsampled z-transform
• Downsampled z-transform
• Downsampled Spectrum
• Power Spectral Density
• Perfect Reconstruction
• Commutators
• Summary
• MATLAB routines
−n
n −n
v[n]z
=
u[
n
n s.t. K|n
K ]z
P
= m u[m]z −Km = U (z K )
V (z) =
P
P
Spectrum: V (ejω ) = U (ejKω )
Spectrum is horizontally shrunk and replicated K times.
1
Total energy unchanged; power (= energy/sample) multiplied by K
Example:
K = 3: three images of the
original spectrum in all.
R
R
1
1
jω 2
jω 2
U (e ) dω = 2π V (e ) dω
Energy unchanged: 2π
1
1
0.5
0.5
0
DSP and Digital Filters (2016-9045)
-2
0
ω
2
0
-2
0
ω
2
Multirate: 11 – 7 / 14
Upsampled z-transform
11: Multirate Systems
• Multirate Systems
• Building blocks
• Resampling Cascades
• Noble Identities
• Noble Identities Proof
• Upsampled z-transform
• Downsampled z-transform
• Downsampled Spectrum
• Power Spectral Density
• Perfect Reconstruction
• Commutators
• Summary
• MATLAB routines
−n
n −n
v[n]z
=
u[
n
n s.t. K|n
K ]z
P
= m u[m]z −Km = U (z K )
V (z) =
P
P
Spectrum: V (ejω ) = U (ejKω )
Spectrum is horizontally shrunk and replicated K times.
1
Total energy unchanged; power (= energy/sample) multiplied by K
Upsampling normally followed by a LP filter to remove images.
Example:
K = 3: three images of the
original spectrum in all.
R
R
1
1
jω 2
jω 2
U (e ) dω = 2π V (e ) dω
Energy unchanged: 2π
1
1
0.5
0.5
0
DSP and Digital Filters (2016-9045)
-2
0
ω
2
0
-2
0
ω
2
Multirate: 11 – 7 / 14
Downsampled z-transform
11: Multirate Systems
• Multirate Systems
• Building blocks
• Resampling Cascades
• Noble Identities
• Noble Identities Proof
• Upsampled z-transform
• Downsampled z-transform
• Downsampled Spectrum
• Power Spectral Density
• Perfect Reconstruction
• Commutators
• Summary
• MATLAB routines
Define cK [n] = δK|n [n]
DSP and Digital Filters (2016-9045)
Multirate: 11 – 8 / 14
Downsampled z-transform
11: Multirate Systems
• Multirate Systems
• Building blocks
• Resampling Cascades
• Noble Identities
• Noble Identities Proof
• Upsampled z-transform
• Downsampled z-transform
• Downsampled Spectrum
• Power Spectral Density
• Perfect Reconstruction
• Commutators
• Summary
• MATLAB routines
PK−1 j2πkn
1
K
Define cK [n] = δK|n [n] = K
k=0 e
DSP and Digital Filters (2016-9045)
Multirate: 11 – 8 / 14
Downsampled z-transform
11: Multirate Systems
• Multirate Systems
• Building blocks
• Resampling Cascades
• Noble Identities
• Noble Identities Proof
• Upsampled z-transform
• Downsampled z-transform
• Downsampled Spectrum
• Power Spectral Density
• Perfect Reconstruction
• Commutators
• Summary
• MATLAB routines
PK−1 j2πkn
1
K
Define cK [n] = δK|n [n] = K
k=0 e
Now define xK [n] =
DSP and Digital Filters (2016-9045)
(
x[n] K | n
0
K∤n
Multirate: 11 – 8 / 14
Downsampled z-transform
11: Multirate Systems
• Multirate Systems
• Building blocks
• Resampling Cascades
• Noble Identities
• Noble Identities Proof
• Upsampled z-transform
• Downsampled z-transform
• Downsampled Spectrum
• Power Spectral Density
• Perfect Reconstruction
• Commutators
• Summary
• MATLAB routines
PK−1 j2πkn
1
K
Define cK [n] = δK|n [n] = K
k=0 e
Now define xK [n] =
DSP and Digital Filters (2016-9045)
(
x[n] K | n
= cK [n]x[n]
0
K∤n
Multirate: 11 – 8 / 14
Downsampled z-transform
11: Multirate Systems
• Multirate Systems
• Building blocks
• Resampling Cascades
• Noble Identities
• Noble Identities Proof
• Upsampled z-transform
• Downsampled z-transform
• Downsampled Spectrum
• Power Spectral Density
• Perfect Reconstruction
• Commutators
• Summary
• MATLAB routines
PK−1 j2πkn
1
K
Define cK [n] = δK|n [n] = K
k=0 e
Now define xK [n] =
XK (z) =
DSP and Digital Filters (2016-9045)
P
n
(
x[n] K | n
= cK [n]x[n]
0
K∤n
xK [n]z −n
Multirate: 11 – 8 / 14
Downsampled z-transform
11: Multirate Systems
• Multirate Systems
• Building blocks
• Resampling Cascades
• Noble Identities
• Noble Identities Proof
• Upsampled z-transform
• Downsampled z-transform
• Downsampled Spectrum
• Power Spectral Density
• Perfect Reconstruction
• Commutators
• Summary
• MATLAB routines
PK−1 j2πkn
1
K
Define cK [n] = δK|n [n] = K
k=0 e
(
x[n] K | n
= cK [n]x[n]
0
K∤n
P PK−1 j2πkn
P
1
−n
XK (z) = n xK [n]z = K n k=0 e K x[n]z −n
Now define xK [n] =
DSP and Digital Filters (2016-9045)
Multirate: 11 – 8 / 14
Downsampled z-transform
11: Multirate Systems
• Multirate Systems
• Building blocks
• Resampling Cascades
• Noble Identities
• Noble Identities Proof
• Upsampled z-transform
• Downsampled z-transform
• Downsampled Spectrum
• Power Spectral Density
• Perfect Reconstruction
• Commutators
• Summary
• MATLAB routines
PK−1 j2πkn
1
K
Define cK [n] = δK|n [n] = K
k=0 e
(
x[n] K | n
= cK [n]x[n]
0
K∤n
P PK−1 j2πkn
P
1
−n
XK (z) = n xK [n]z = K n k=0 e K x[n]z −n
−j2πk −n
P
P
K−1
1
K
=K
z
k=0
n x[n] e
Now define xK [n] =
DSP and Digital Filters (2016-9045)
Multirate: 11 – 8 / 14
Downsampled z-transform
11: Multirate Systems
• Multirate Systems
• Building blocks
• Resampling Cascades
• Noble Identities
• Noble Identities Proof
• Upsampled z-transform
• Downsampled z-transform
• Downsampled Spectrum
• Power Spectral Density
• Perfect Reconstruction
• Commutators
• Summary
• MATLAB routines
PK−1 j2πkn
1
K
Define cK [n] = δK|n [n] = K
k=0 e
(
x[n] K | n
= cK [n]x[n]
0
K∤n
P PK−1 j2πkn
P
1
−n
XK (z) = n xK [n]z = K n k=0 e K x[n]z −n
−j2πk −n
P
PK−1
P
−j2πk
K−1
1
1
K
= K k=0
z
= K k=0 X(e K z)
n x[n] e
Now define xK [n] =
DSP and Digital Filters (2016-9045)
Multirate: 11 – 8 / 14
Downsampled z-transform
11: Multirate Systems
• Multirate Systems
• Building blocks
• Resampling Cascades
• Noble Identities
• Noble Identities Proof
• Upsampled z-transform
• Downsampled z-transform
• Downsampled Spectrum
• Power Spectral Density
• Perfect Reconstruction
• Commutators
• Summary
• MATLAB routines
PK−1 j2πkn
1
K
Define cK [n] = δK|n [n] = K
k=0 e
(
x[n] K | n
= cK [n]x[n]
0
K∤n
P PK−1 j2πkn
P
1
−n
XK (z) = n xK [n]z = K n k=0 e K x[n]z −n
−j2πk −n
P
PK−1
P
−j2πk
K−1
1
1
K
= K k=0
z
= K k=0 X(e K z)
n x[n] e
Now define xK [n] =
From previous slide:
XK (z) = Y (z K )
DSP and Digital Filters (2016-9045)
Multirate: 11 – 8 / 14
Downsampled z-transform
11: Multirate Systems
• Multirate Systems
• Building blocks
• Resampling Cascades
• Noble Identities
• Noble Identities Proof
• Upsampled z-transform
• Downsampled z-transform
• Downsampled Spectrum
• Power Spectral Density
• Perfect Reconstruction
• Commutators
• Summary
• MATLAB routines
PK−1 j2πkn
1
K
Define cK [n] = δK|n [n] = K
k=0 e
(
x[n] K | n
= cK [n]x[n]
0
K∤n
P PK−1 j2πkn
P
1
−n
XK (z) = n xK [n]z = K n k=0 e K x[n]z −n
−j2πk −n
P
PK−1
P
−j2πk
K−1
1
1
K
= K k=0
z
= K k=0 X(e K z)
n x[n] e
Now define xK [n] =
From previous slide:
XK (z) = Y (z K )
1
⇒ Y (z) = XK (z K )
DSP and Digital Filters (2016-9045)
Multirate: 11 – 8 / 14
Downsampled z-transform
11: Multirate Systems
• Multirate Systems
• Building blocks
• Resampling Cascades
• Noble Identities
• Noble Identities Proof
• Upsampled z-transform
• Downsampled z-transform
• Downsampled Spectrum
• Power Spectral Density
• Perfect Reconstruction
• Commutators
• Summary
• MATLAB routines
PK−1 j2πkn
1
K
Define cK [n] = δK|n [n] = K
k=0 e
(
x[n] K | n
= cK [n]x[n]
0
K∤n
P PK−1 j2πkn
P
1
−n
XK (z) = n xK [n]z = K n k=0 e K x[n]z −n
−j2πk −n
P
PK−1
P
−j2πk
K−1
1
1
K
= K k=0
z
= K k=0 X(e K z)
n x[n] e
Now define xK [n] =
From previous slide:
XK (z) = Y (z K )
1
⇒ Y (z) = XK (z K ) =
DSP and Digital Filters (2016-9045)
1
K
PK−1
k=0
X(e
−j2πk
K
1
zK)
Multirate: 11 – 8 / 14
Downsampled z-transform
11: Multirate Systems
• Multirate Systems
• Building blocks
• Resampling Cascades
• Noble Identities
• Noble Identities Proof
• Upsampled z-transform
• Downsampled z-transform
• Downsampled Spectrum
• Power Spectral Density
• Perfect Reconstruction
• Commutators
• Summary
• MATLAB routines
PK−1 j2πkn
1
K
Define cK [n] = δK|n [n] = K
k=0 e
(
x[n] K | n
= cK [n]x[n]
0
K∤n
P PK−1 j2πkn
P
1
−n
XK (z) = n xK [n]z = K n k=0 e K x[n]z −n
−j2πk −n
P
PK−1
P
−j2πk
K−1
1
1
K
= K k=0
z
= K k=0 X(e K z)
n x[n] e
Now define xK [n] =
From previous slide:
XK (z) = Y (z K )
1
⇒ Y (z) = XK (z K ) =
1
K
Frequency Spectrum:
Y (ejω ) =
DSP and Digital Filters (2016-9045)
1
K
PK−1
k=0
X(e
PK−1
k=0
j(ω−2πk)
K
X(e
−j2πk
K
1
zK)
)
Multirate: 11 – 8 / 14
Downsampled z-transform
11: Multirate Systems
• Multirate Systems
• Building blocks
• Resampling Cascades
• Noble Identities
• Noble Identities Proof
• Upsampled z-transform
• Downsampled z-transform
• Downsampled Spectrum
• Power Spectral Density
• Perfect Reconstruction
• Commutators
• Summary
• MATLAB routines
PK−1 j2πkn
1
K
Define cK [n] = δK|n [n] = K
k=0 e
(
x[n] K | n
= cK [n]x[n]
0
K∤n
P PK−1 j2πkn
P
1
−n
XK (z) = n xK [n]z = K n k=0 e K x[n]z −n
−j2πk −n
P
PK−1
P
−j2πk
K−1
1
1
K
= K k=0
z
= K k=0 X(e K z)
n x[n] e
Now define xK [n] =
From previous slide:
XK (z) = Y (z K )
1
⇒ Y (z) = XK (z K ) =
Frequency Spectrum:
Y (ejω ) =
=
DSP and Digital Filters (2016-9045)
1
K
1
K
1
K
PK−1
k=0
X(e
−j2πk
K
1
zK)
PK−1
j(ω−2πk)
K
)
X(e
k=0jω
jω
jω
2π
4π
X(e K ) + X(e K − K ) + X(e K − K ) + · · ·
Multirate: 11 – 8 / 14
Downsampled z-transform
11: Multirate Systems
• Multirate Systems
• Building blocks
• Resampling Cascades
• Noble Identities
• Noble Identities Proof
• Upsampled z-transform
• Downsampled z-transform
• Downsampled Spectrum
• Power Spectral Density
• Perfect Reconstruction
• Commutators
• Summary
• MATLAB routines
PK−1 j2πkn
1
K
Define cK [n] = δK|n [n] = K
k=0 e
(
x[n] K | n
= cK [n]x[n]
0
K∤n
P PK−1 j2πkn
P
1
−n
XK (z) = n xK [n]z = K n k=0 e K x[n]z −n
−j2πk −n
P
PK−1
P
−j2πk
K−1
1
1
K
= K k=0
z
= K k=0 X(e K z)
n x[n] e
Now define xK [n] =
From previous slide:
XK (z) = Y (z K )
1
⇒ Y (z) = XK (z K ) =
Frequency Spectrum:
1
K
PK−1
k=0
X(e
−j2πk
K
1
zK)
PK−1
j(ω−2πk)
K
)
X(e
k=0jω
jω
jω
2π
4π
=
X(e K ) + X(e K − K ) + X(e K − K ) + · · ·
Average of K aliased versions, each expanded in ω by a factor of K .
Y (ejω ) =
DSP and Digital Filters (2016-9045)
1
K
1
K
Multirate: 11 – 8 / 14
Downsampled z-transform
11: Multirate Systems
• Multirate Systems
• Building blocks
• Resampling Cascades
• Noble Identities
• Noble Identities Proof
• Upsampled z-transform
• Downsampled z-transform
• Downsampled Spectrum
• Power Spectral Density
• Perfect Reconstruction
• Commutators
• Summary
• MATLAB routines
PK−1 j2πkn
1
K
Define cK [n] = δK|n [n] = K
k=0 e
(
x[n] K | n
= cK [n]x[n]
0
K∤n
P PK−1 j2πkn
P
1
−n
XK (z) = n xK [n]z = K n k=0 e K x[n]z −n
−j2πk −n
P
PK−1
P
−j2πk
K−1
1
1
K
= K k=0
z
= K k=0 X(e K z)
n x[n] e
Now define xK [n] =
From previous slide:
XK (z) = Y (z K )
1
⇒ Y (z) = XK (z K ) =
Frequency Spectrum:
1
K
PK−1
k=0
X(e
−j2πk
K
1
zK)
PK−1
j(ω−2πk)
K
)
X(e
k=0jω
jω
jω
2π
4π
=
X(e K ) + X(e K − K ) + X(e K − K ) + · · ·
Average of K aliased versions, each expanded in ω by a factor of K .
Y (ejω ) =
1
K
1
K
Downsampling is normally preceded by a LP filter to prevent aliasing.
DSP and Digital Filters (2016-9045)
Multirate: 11 – 8 / 14
Downsampled Spectrum
11: Multirate Systems
• Multirate Systems
• Building blocks
• Resampling Cascades
• Noble Identities
• Noble Identities Proof
• Upsampled z-transform
• Downsampled z-transform
• Downsampled Spectrum
• Power Spectral Density
• Perfect Reconstruction
• Commutators
• Summary
• MATLAB routines
jω
Y (e ) =
DSP and Digital Filters (2016-9045)
1
K
PK−1
k=0
X(e
j(ω−2πk)
K
)
Multirate: 11 – 9 / 14
Downsampled Spectrum
11: Multirate Systems
• Multirate Systems
• Building blocks
• Resampling Cascades
• Noble Identities
• Noble Identities Proof
• Upsampled z-transform
• Downsampled z-transform
• Downsampled Spectrum
• Power Spectral Density
• Perfect Reconstruction
• Commutators
• Summary
• MATLAB routines
jω
Y (e ) =
Example 1:
1
K
PK−1
k=0
X(e
K=3
π
Not quite limited to ± K
DSP and Digital Filters (2016-9045)
j(ω−2πk)
K
)
1
0.5
0
-2
0
ω
2
Multirate: 11 – 9 / 14
Downsampled Spectrum
11: Multirate Systems
• Multirate Systems
• Building blocks
• Resampling Cascades
• Noble Identities
• Noble Identities Proof
• Upsampled z-transform
• Downsampled z-transform
• Downsampled Spectrum
• Power Spectral Density
• Perfect Reconstruction
• Commutators
• Summary
• MATLAB routines
jω
Y (e ) =
Example 1:
1
K
PK−1
k=0
X(e
j(ω−2πk)
K
K=3
π
Not quite limited to ± K
Shaded region shows aliasing
DSP and Digital Filters (2016-9045)
)
1
1
0.5
0.5
0
-2
0
ω
2
0
-2
0
ω
2
Multirate: 11 – 9 / 14
Downsampled Spectrum
11: Multirate Systems
• Multirate Systems
• Building blocks
• Resampling Cascades
• Noble Identities
• Noble Identities Proof
• Upsampled z-transform
• Downsampled z-transform
• Downsampled Spectrum
• Power Spectral Density
• Perfect Reconstruction
• Commutators
• Summary
• MATLAB routines
jω
Y (e ) =
Example 1:
1
K
PK−1
k=0
X(e
j(ω−2πk)
K
K=3
π
Not quite limited to ± K
Shaded region shows aliasing
Energy decreases:
DSP and Digital Filters (2016-9045)
1
2π
)
1
1
0.5
0.5
0
-2
R
Y (ejω )2 dω ≈
0
ω
1
K
2
×
1
2π
0
-2
0
ω
2
R
X(ejω )2 dω
Multirate: 11 – 9 / 14
Downsampled Spectrum
11: Multirate Systems
• Multirate Systems
• Building blocks
• Resampling Cascades
• Noble Identities
• Noble Identities Proof
• Upsampled z-transform
• Downsampled z-transform
• Downsampled Spectrum
• Power Spectral Density
• Perfect Reconstruction
• Commutators
• Summary
• MATLAB routines
jω
Y (e ) =
Example 1:
1
K
PK−1
k=0
X(e
j(ω−2πk)
K
K=3
π
Not quite limited to ± K
Shaded region shows aliasing
Energy decreases:
Example 2:
K=3
1
2π
)
1
1
0.5
0.5
0
R
Y (ejω )2 dω ≈
π
π
≤ |ω| < 2 K
Energy all in K
0
ω
1
K
2
×
1
2π
1
0
-2
0
ω
2
R
X(ejω )2 dω
0.5
0
DSP and Digital Filters (2016-9045)
-2
-2
0
ω
2
Multirate: 11 – 9 / 14
Downsampled Spectrum
11: Multirate Systems
• Multirate Systems
• Building blocks
• Resampling Cascades
• Noble Identities
• Noble Identities Proof
• Upsampled z-transform
• Downsampled z-transform
• Downsampled Spectrum
• Power Spectral Density
• Perfect Reconstruction
• Commutators
• Summary
• MATLAB routines
jω
Y (e ) =
Example 1:
1
K
PK−1
k=0
X(e
j(ω−2πk)
K
K=3
π
Not quite limited to ± K
Shaded region shows aliasing
Energy decreases:
Example 2:
K=3
1
2π
1
1
0.5
0.5
0
-2
R
Y (ejω )2 dω ≈
π
π
≤ |ω| < 2 K
Energy all in K
No aliasing: ,
DSP and Digital Filters (2016-9045)
)
0
ω
1
K
2
×
1
2π
1
-2
0
ω
2
R
X(ejω )2 dω
1
0.5
0
0
0.5
-2
0
ω
2
0
-2
0
ω
2
Multirate: 11 – 9 / 14
Downsampled Spectrum
11: Multirate Systems
• Multirate Systems
• Building blocks
• Resampling Cascades
• Noble Identities
• Noble Identities Proof
• Upsampled z-transform
• Downsampled z-transform
• Downsampled Spectrum
• Power Spectral Density
• Perfect Reconstruction
• Commutators
• Summary
• MATLAB routines
jω
Y (e ) =
Example 1:
1
K
PK−1
k=0
X(e
j(ω−2πk)
K
K=3
π
Not quite limited to ± K
Shaded region shows aliasing
Energy decreases:
Example 2:
K=3
1
2π
)
1
1
0.5
0.5
0
-2
R
Y (ejω )2 dω ≈
π
π
≤ |ω| < 2 K
Energy all in K
No aliasing: ,
0
ω
1
K
2
×
1
2π
1
-2
0
ω
2
R
X(ejω )2 dω
1
0.5
0
0
0.5
-2
0
ω
2
0
-2
0
ω
2
π
π
No aliasing: If all energy is in r K
≤ |ω| < (r + 1) K
for some integer r
DSP and Digital Filters (2016-9045)
Multirate: 11 – 9 / 14
Downsampled Spectrum
11: Multirate Systems
• Multirate Systems
• Building blocks
• Resampling Cascades
• Noble Identities
• Noble Identities Proof
• Upsampled z-transform
• Downsampled z-transform
• Downsampled Spectrum
• Power Spectral Density
• Perfect Reconstruction
• Commutators
• Summary
• MATLAB routines
jω
Y (e ) =
Example 1:
1
K
PK−1
k=0
X(e
j(ω−2πk)
K
K=3
π
Not quite limited to ± K
Shaded region shows aliasing
Energy decreases:
Example 2:
K=3
1
2π
)
1
1
0.5
0.5
0
-2
R
Y (ejω )2 dω ≈
π
π
≤ |ω| < 2 K
Energy all in K
No aliasing: ,
0
ω
1
K
2
×
1
2π
1
-2
0
ω
2
R
X(ejω )2 dω
1
0.5
0
0
0.5
-2
0
ω
2
0
-2
0
ω
2
π
π
No aliasing: If all energy is in r K
≤ |ω| < (r + 1) K
for some integer r
π
Normal case (r = 0): If all energy in 0 ≤ |ω| ≤ K
DSP and Digital Filters (2016-9045)
Multirate: 11 – 9 / 14
Downsampled Spectrum
11: Multirate Systems
• Multirate Systems
• Building blocks
• Resampling Cascades
• Noble Identities
• Noble Identities Proof
• Upsampled z-transform
• Downsampled z-transform
• Downsampled Spectrum
• Power Spectral Density
• Perfect Reconstruction
• Commutators
• Summary
• MATLAB routines
jω
Y (e ) =
Example 1:
1
K
PK−1
k=0
X(e
j(ω−2πk)
K
K=3
π
Not quite limited to ± K
Shaded region shows aliasing
Energy decreases:
Example 2:
K=3
1
2π
)
1
1
0.5
0.5
0
-2
R
Y (ejω )2 dω ≈
π
π
≤ |ω| < 2 K
Energy all in K
No aliasing: ,
0
ω
1
K
2
×
1
2π
1
-2
0
ω
2
R
X(ejω )2 dω
1
0.5
0
0
0.5
-2
0
ω
2
0
-2
0
ω
2
π
π
No aliasing: If all energy is in r K
≤ |ω| < (r + 1) K
for some integer r
π
Normal case (r = 0): If all energy in 0 ≤ |ω| ≤ K
1
1
Downsampling: Total energy multiplied by ≈ K
(= K
if no aliasing)
Average power ≈ unchanged (= energy/sample)
DSP and Digital Filters (2016-9045)
Multirate: 11 – 9 / 14
Power Spectral Density
• Multirate Systems
• Building blocks
• Resampling Cascades
• Noble Identities
• Noble Identities Proof
• Upsampled z-transform
• Downsampled z-transform
• Downsampled Spectrum
• Power Spectral Density
• Perfect Reconstruction
• Commutators
• Summary
• MATLAB routines
Example: Signal in ω ∈ ±0.4π + Tone @ ω = ±0.1π + White noise
DSP and Digital Filters (2016-9045)
original rate
PSD , ∫ = 0.5 + 0.1 = 0.6
11: Multirate Systems
1
0.5
0
-3
-2
-1
0
1
Frequency (rad/samp)
2
3
Multirate: 11 – 10 / 14
Power Spectral Density
• Multirate Systems
• Building blocks
• Resampling Cascades
• Noble Identities
• Noble Identities Proof
• Upsampled z-transform
• Downsampled z-transform
• Downsampled Spectrum
• Power Spectral Density
• Perfect Reconstruction
• Commutators
• Summary
• MATLAB routines
Example: Signal in ω ∈ ±0.4π + Tone @ ω = ±0.1π + White noise
Power = Energy/sample = Average PSD
DSP and Digital Filters (2016-9045)
original rate
PSD , ∫ = 0.5 + 0.1 = 0.6
11: Multirate Systems
1
0.5
0
-3
-2
-1
0
1
Frequency (rad/samp)
2
3
Multirate: 11 – 10 / 14
Power Spectral Density
• Multirate Systems
• Building blocks
• Resampling Cascades
• Noble Identities
• Noble Identities Proof
• Upsampled z-transform
• Downsampled z-transform
• Downsampled Spectrum
• Power Spectral Density
• Perfect Reconstruction
• Commutators
• Summary
• MATLAB routines
Example: Signal in ω ∈ ±0.4π + Tone @ ω = ±0.1π + White noise
Power = Energy/sample = Average PSD
DSP and Digital Filters (2016-9045)
=
1
2π
Rπ
−π
PSD(ω)dω
original rate
PSD , ∫ = 0.5 + 0.1 = 0.6
11: Multirate Systems
1
0.5
0
-3
-2
-1
0
1
Frequency (rad/samp)
2
3
Multirate: 11 – 10 / 14
Power Spectral Density
• Multirate Systems
• Building blocks
• Resampling Cascades
• Noble Identities
• Noble Identities Proof
• Upsampled z-transform
• Downsampled z-transform
• Downsampled Spectrum
• Power Spectral Density
• Perfect Reconstruction
• Commutators
• Summary
• MATLAB routines
Example: Signal in ω ∈ ±0.4π + Tone @ ω = ±0.1π + White noise
Power = Energy/sample = Average PSD
DSP and Digital Filters (2016-9045)
=
1
2π
Rπ
−π
PSD(ω)dω = 0.6
original rate
PSD , ∫ = 0.5 + 0.1 = 0.6
11: Multirate Systems
1
0.5
0
-3
-2
-1
0
1
Frequency (rad/samp)
2
3
Multirate: 11 – 10 / 14
Power Spectral Density
• Multirate Systems
• Building blocks
• Resampling Cascades
• Noble Identities
• Noble Identities Proof
• Upsampled z-transform
• Downsampled z-transform
• Downsampled Spectrum
• Power Spectral Density
• Perfect Reconstruction
• Commutators
• Summary
• MATLAB routines
Example: Signal in ω ∈ ±0.4π + Tone @ ω = ±0.1π + White noise
Power = Energy/sample = Average PSD
=
1
2π
Rπ
−π
PSD(ω)dω = 0.6
Component powers:
Signal = 0.3, Tone = 0.2, Noise = 0.1
DSP and Digital Filters (2016-9045)
original rate
PSD , ∫ = 0.5 + 0.1 = 0.6
11: Multirate Systems
1
0.5
0
-3
-2
-1
0
1
Frequency (rad/samp)
2
3
Multirate: 11 – 10 / 14
Power Spectral Density
Example: Signal in ω ∈ ±0.4π + Tone @ ω = ±0.1π + White noise
Power = Energy/sample = Average PSD
=
1
2π
Rπ
−π
original rate
PSD , ∫ = 0.5 + 0.1 = 0.6
• Multirate Systems
• Building blocks
• Resampling Cascades
• Noble Identities
• Noble Identities Proof
• Upsampled z-transform
• Downsampled z-transform
• Downsampled Spectrum
• Power Spectral Density
• Perfect Reconstruction
• Commutators
• Summary
• MATLAB routines
PSD(ω)dω = 0.6
Component powers:
Signal = 0.3, Tone = 0.2, Noise = 0.1
1
0.5
0
-3
-2
-1
0
1
Frequency (rad/samp)
2
3
Upsampling:
Same energy
per second
⇒ Power is ÷K
DSP and Digital Filters (2016-9045)
PSD , ∫ = 0.13 + 0.18 = 0.3
11: Multirate Systems
upsample × 2
0.4
0.2
0
-3
-2
-1
0
1
Frequency (rad/samp)
2
3
Multirate: 11 – 10 / 14
Power Spectral Density
Example: Signal in ω ∈ ±0.4π + Tone @ ω = ±0.1π + White noise
Power = Energy/sample = Average PSD
Rπ
−π
PSD , ∫ = 0.5 + 0.1 = 0.6
=
1
2π
original rate
PSD(ω)dω = 0.6
Component powers:
Signal = 0.3, Tone = 0.2, Noise = 0.1
1
0.5
0
-3
-2
-1
0
1
Frequency (rad/samp)
2
3
2
3
Upsampling:
Same energy
per second
⇒ Power is ÷K
DSP and Digital Filters (2016-9045)
upsample × 2
0.4
0.2
0
-3
-2
-1
0
1
Frequency (rad/samp)
2
3
PSD , ∫ = 0.056 + 0.14 = 0.2
• Multirate Systems
• Building blocks
• Resampling Cascades
• Noble Identities
• Noble Identities Proof
• Upsampled z-transform
• Downsampled z-transform
• Downsampled Spectrum
• Power Spectral Density
• Perfect Reconstruction
• Commutators
• Summary
• MATLAB routines
PSD , ∫ = 0.13 + 0.18 = 0.3
11: Multirate Systems
upsample × 3
0.3
0.2
0.1
0
-3
-2
-1
0
1
Frequency (rad/samp)
Multirate: 11 – 10 / 14
Power Spectral Density
Example: Signal in ω ∈ ±0.4π + Tone @ ω = ±0.1π + White noise
Power = Energy/sample = Average PSD
Rπ
−π
PSD , ∫ = 0.5 + 0.1 = 0.6
=
1
2π
original rate
PSD(ω)dω = 0.6
Component powers:
Signal = 0.3, Tone = 0.2, Noise = 0.1
1
0.5
0
-3
-2
-1
0
1
Frequency (rad/samp)
2
3
2
3
Upsampling:
Same energy
per second
⇒ Power is ÷K
upsample × 2
0.4
0.2
0
-3
-2
-1
0
1
Frequency (rad/samp)
2
3
2
3
PSD , ∫ = 0.056 + 0.14 = 0.2
• Multirate Systems
• Building blocks
• Resampling Cascades
• Noble Identities
• Noble Identities Proof
• Upsampled z-transform
• Downsampled z-transform
• Downsampled Spectrum
• Power Spectral Density
• Perfect Reconstruction
• Commutators
• Summary
• MATLAB routines
PSD , ∫ = 0.13 + 0.18 = 0.3
11: Multirate Systems
upsample × 3
0.3
0.2
0.1
0
-3
-2
-1
0
1
Frequency (rad/samp)
Average power
is unchanged.
DSP and Digital Filters (2016-9045)
PSD , ∫ = 0.5 + 0.1 = 0.6
Downsampling:
downsample ÷ 2
0.6
0.4
0.2
0
-3
-2
-1
0
1
Frequency (rad/samp)
Multirate: 11 – 10 / 14
Power Spectral Density
Example: Signal in ω ∈ ±0.4π + Tone @ ω = ±0.1π + White noise
Power = Energy/sample = Average PSD
Rπ
−π
PSD , ∫ = 0.5 + 0.1 = 0.6
=
1
2π
original rate
PSD(ω)dω = 0.6
Component powers:
Signal = 0.3, Tone = 0.2, Noise = 0.1
1
0.5
0
-3
-2
-1
0
1
Frequency (rad/samp)
2
3
2
3
2
3
Upsampling:
Same energy
per second
⇒ Power is ÷K
upsample × 2
0.4
0.2
0
-3
-2
-1
0
1
Frequency (rad/samp)
2
3
PSD , ∫ = 0.056 + 0.14 = 0.2
• Multirate Systems
• Building blocks
• Resampling Cascades
• Noble Identities
• Noble Identities Proof
• Upsampled z-transform
• Downsampled z-transform
• Downsampled Spectrum
• Power Spectral Density
• Perfect Reconstruction
• Commutators
• Summary
• MATLAB routines
PSD , ∫ = 0.13 + 0.18 = 0.3
11: Multirate Systems
upsample × 3
0.3
0.2
0.1
0
-3
-2
-1
0
1
Frequency (rad/samp)
DSP and Digital Filters (2016-9045)
downsample ÷ 2
0.6
0.4
0.2
0
-3
-2
-1
0
1
Frequency (rad/samp)
2
3
PSD , ∫ = 0.49 + 0.11 = 0.6
Average power
is unchanged.
∃ aliasing in
the ÷3 case.
PSD , ∫ = 0.5 + 0.1 = 0.6
Downsampling:
downsample ÷ 3
0.6
0.4
0.2
0
-3
-2
-1
0
1
Frequency (rad/samp)
Multirate: 11 – 10 / 14
Perfect Reconstruction
11: Multirate Systems
• Multirate Systems
• Building blocks
• Resampling Cascades
• Noble Identities
• Noble Identities Proof
• Upsampled z-transform
• Downsampled z-transform
• Downsampled Spectrum
• Power Spectral Density
• Perfect Reconstruction
• Commutators
• Summary
• MATLAB routines
x[n]
DSP and Digital Filters (2016-9045)
defghijklmn
Multirate: 11 – 11 / 14
Perfect Reconstruction
11: Multirate Systems
• Multirate Systems
• Building blocks
• Resampling Cascades
• Noble Identities
• Noble Identities Proof
• Upsampled z-transform
• Downsampled z-transform
• Downsampled Spectrum
• Power Spectral Density
• Perfect Reconstruction
• Commutators
• Summary
• MATLAB routines
x[n]
u[m]
DSP and Digital Filters (2016-9045)
defghijklmn
f i l
Multirate: 11 – 11 / 14
Perfect Reconstruction
11: Multirate Systems
• Multirate Systems
• Building blocks
• Resampling Cascades
• Noble Identities
• Noble Identities Proof
• Upsampled z-transform
• Downsampled z-transform
• Downsampled Spectrum
• Power Spectral Density
• Perfect Reconstruction
• Commutators
• Summary
• MATLAB routines
x[n]
u[m]
p[n]
DSP and Digital Filters (2016-9045)
defghijklmn
f i l
---f--i--l
Multirate: 11 – 11 / 14
Perfect Reconstruction
11: Multirate Systems
• Multirate Systems
• Building blocks
• Resampling Cascades
• Noble Identities
• Noble Identities Proof
• Upsampled z-transform
• Downsampled z-transform
• Downsampled Spectrum
• Power Spectral Density
• Perfect Reconstruction
• Commutators
• Summary
• MATLAB routines
x[n]
u[m]
p[n]
v[m]
DSP and Digital Filters (2016-9045)
defghijklmn
f i l
---f--i--l
b e h k
Multirate: 11 – 11 / 14
Perfect Reconstruction
11: Multirate Systems
• Multirate Systems
• Building blocks
• Resampling Cascades
• Noble Identities
• Noble Identities Proof
• Upsampled z-transform
• Downsampled z-transform
• Downsampled Spectrum
• Power Spectral Density
• Perfect Reconstruction
• Commutators
• Summary
• MATLAB routines
x[n]
u[m]
p[n]
v[m]
q[n]
DSP and Digital Filters (2016-9045)
defghijklmn
f i l
---f--i--l
b e h k
-b-ef-hi-kl
Multirate: 11 – 11 / 14
Perfect Reconstruction
11: Multirate Systems
• Multirate Systems
• Building blocks
• Resampling Cascades
• Noble Identities
• Noble Identities Proof
• Upsampled z-transform
• Downsampled z-transform
• Downsampled Spectrum
• Power Spectral Density
• Perfect Reconstruction
• Commutators
• Summary
• MATLAB routines
x[n]
u[m]
p[n]
v[m]
q[n]
w[m]
DSP and Digital Filters (2016-9045)
defghijklmn
f i l
---f--i--l
b e h k
-b-ef-hi-kl
a d g j
Multirate: 11 – 11 / 14
Perfect Reconstruction
11: Multirate Systems
• Multirate Systems
• Building blocks
• Resampling Cascades
• Noble Identities
• Noble Identities Proof
• Upsampled z-transform
• Downsampled z-transform
• Downsampled Spectrum
• Power Spectral Density
• Perfect Reconstruction
• Commutators
• Summary
• MATLAB routines
x[n]
u[m]
p[n]
v[m]
q[n]
w[m]
y[n]
DSP and Digital Filters (2016-9045)
defghijklmn
f i l
---f--i--l
b e h k
-b-ef-hi-kl
a d g j
abdefghijkl
Multirate: 11 – 11 / 14
Perfect Reconstruction
11: Multirate Systems
• Multirate Systems
• Building blocks
• Resampling Cascades
• Noble Identities
• Noble Identities Proof
• Upsampled z-transform
• Downsampled z-transform
• Downsampled Spectrum
• Power Spectral Density
• Perfect Reconstruction
• Commutators
• Summary
• MATLAB routines
x[n]
u[m]
p[n]
v[m]
q[n]
w[m]
y[n]
defghijklmn
f i l
---f--i--l
b e h k
-b-ef-hi-kl
a d g j
abdefghijkl
Input sequence x[n] is split into three streams at 13 the sample rate:
u[m] = x[3m], v[m] = x[3m − 1], w[m] = x[3m − 2]
DSP and Digital Filters (2016-9045)
Multirate: 11 – 11 / 14
Perfect Reconstruction
11: Multirate Systems
• Multirate Systems
• Building blocks
• Resampling Cascades
• Noble Identities
• Noble Identities Proof
• Upsampled z-transform
• Downsampled z-transform
• Downsampled Spectrum
• Power Spectral Density
• Perfect Reconstruction
• Commutators
• Summary
• MATLAB routines
x[n]
u[m]
p[n]
v[m]
q[n]
w[m]
y[n]
defghijklmn
f i l
---f--i--l
b e h k
-b-ef-hi-kl
a d g j
abdefghijkl
Input sequence x[n] is split into three streams at 13 the sample rate:
u[m] = x[3m], v[m] = x[3m − 1], w[m] = x[3m − 2]
Following upsampling, the streams are aligned by the delays and then
added to give:
y[n] = x[n − 2]
DSP and Digital Filters (2016-9045)
Multirate: 11 – 11 / 14
Perfect Reconstruction
11: Multirate Systems
• Multirate Systems
• Building blocks
• Resampling Cascades
• Noble Identities
• Noble Identities Proof
• Upsampled z-transform
• Downsampled z-transform
• Downsampled Spectrum
• Power Spectral Density
• Perfect Reconstruction
• Commutators
• Summary
• MATLAB routines
x[n]
u[m]
p[n]
v[m]
q[n]
w[m]
y[n]
defghijklmn
f i l
---f--i--l
b e h k
-b-ef-hi-kl
a d g j
abdefghijkl
Input sequence x[n] is split into three streams at 13 the sample rate:
u[m] = x[3m], v[m] = x[3m − 1], w[m] = x[3m − 2]
Following upsampling, the streams are aligned by the delays and then
added to give:
y[n] = x[n − 2]
Perfect Reconstruction: output is a delayed scaled replica of the input
DSP and Digital Filters (2016-9045)
Multirate: 11 – 11 / 14
Commutators
11: Multirate Systems
• Multirate Systems
• Building blocks
• Resampling Cascades
• Noble Identities
• Noble Identities Proof
• Upsampled z-transform
• Downsampled z-transform
• Downsampled Spectrum
• Power Spectral Density
• Perfect Reconstruction
• Commutators
• Summary
• MATLAB routines
DSP and Digital Filters (2016-9045)
x[n]
u[m]
v[m]
w[m]
defghijklmn
f i l
b e h k
a d g j
y[n]
abdefghijkl
Multirate: 11 – 12 / 14
Commutators
11: Multirate Systems
• Multirate Systems
• Building blocks
• Resampling Cascades
• Noble Identities
• Noble Identities Proof
• Upsampled z-transform
• Downsampled z-transform
• Downsampled Spectrum
• Power Spectral Density
• Perfect Reconstruction
• Commutators
• Summary
• MATLAB routines
x[n]
u[m]
v[m]
w[m]
defghijklmn
f i l
b e h k
a d g j
y[n]
abdefghijkl
The combination of delays and downsamplers can be regarded as a
commutator that distributes values in sequence to u, w and v .
DSP and Digital Filters (2016-9045)
Multirate: 11 – 12 / 14
Commutators
11: Multirate Systems
• Multirate Systems
• Building blocks
• Resampling Cascades
• Noble Identities
• Noble Identities Proof
• Upsampled z-transform
• Downsampled z-transform
• Downsampled Spectrum
• Power Spectral Density
• Perfect Reconstruction
• Commutators
• Summary
• MATLAB routines
x[n]
u[m]
v[m]
w[m]
defghijklmn
f i l
b e h k
a d g j
y[n]
abdefghijkl
The combination of delays and downsamplers can be regarded as a
commutator that distributes values in sequence to u, w and v .
2
1
Fractional delays, z − 3 and z − 3 are needed to synchronize the streams.
DSP and Digital Filters (2016-9045)
Multirate: 11 – 12 / 14
Commutators
11: Multirate Systems
• Multirate Systems
• Building blocks
• Resampling Cascades
• Noble Identities
• Noble Identities Proof
• Upsampled z-transform
• Downsampled z-transform
• Downsampled Spectrum
• Power Spectral Density
• Perfect Reconstruction
• Commutators
• Summary
• MATLAB routines
x[n]
u[m]
v[m]
w[m]
defghijklmn
f i l
b e h k
a d g j
y[n]
abdefghijkl
The combination of delays and downsamplers can be regarded as a
commutator that distributes values in sequence to u, w and v .
2
1
Fractional delays, z − 3 and z − 3 are needed to synchronize the streams.
The output commutator takes values from the streams in sequence.
DSP and Digital Filters (2016-9045)
Multirate: 11 – 12 / 14
Commutators
11: Multirate Systems
• Multirate Systems
• Building blocks
• Resampling Cascades
• Noble Identities
• Noble Identities Proof
• Upsampled z-transform
• Downsampled z-transform
• Downsampled Spectrum
• Power Spectral Density
• Perfect Reconstruction
• Commutators
• Summary
• MATLAB routines
x[n]
u[m]
v[m]
w[m]
v[m + 13 ]
w[m + 23 ]
y[n]
defghijklmn
f i l
b e h k
a d g j
e h k l
d g j m
abdefghijkl
The combination of delays and downsamplers can be regarded as a
commutator that distributes values in sequence to u, w and v .
2
1
Fractional delays, z − 3 and z − 3 are needed to synchronize the streams.
The output commutator takes values from the streams in sequence.
For clarity, we omit the fractional delays and regard each terminal, ◦, as
holding its value until needed.
DSP and Digital Filters (2016-9045)
Multirate: 11 – 12 / 14
Commutators
11: Multirate Systems
• Multirate Systems
• Building blocks
• Resampling Cascades
• Noble Identities
• Noble Identities Proof
• Upsampled z-transform
• Downsampled z-transform
• Downsampled Spectrum
• Power Spectral Density
• Perfect Reconstruction
• Commutators
• Summary
• MATLAB routines
x[n]
u[m]
v[m]
w[m]
v[m + 13 ]
w[m + 23 ]
y[n]
defghijklmn
f i l
b e h k
a d g j
e h k l
d g j m
abdefghijkl
The combination of delays and downsamplers can be regarded as a
commutator that distributes values in sequence to u, w and v .
2
1
Fractional delays, z − 3 and z − 3 are needed to synchronize the streams.
The output commutator takes values from the streams in sequence.
For clarity, we omit the fractional delays and regard each terminal, ◦, as
holding its value until needed. Initial commutator position has zero delay.
DSP and Digital Filters (2016-9045)
Multirate: 11 – 12 / 14
Commutators
11: Multirate Systems
• Multirate Systems
• Building blocks
• Resampling Cascades
• Noble Identities
• Noble Identities Proof
• Upsampled z-transform
• Downsampled z-transform
• Downsampled Spectrum
• Power Spectral Density
• Perfect Reconstruction
• Commutators
• Summary
• MATLAB routines
x[n]
u[m]
v[m]
w[m]
v[m + 13 ]
w[m + 23 ]
y[n]
defghijklmn
f i l
b e h k
a d g j
e h k l
d g j m
abdefghijkl
The combination of delays and downsamplers can be regarded as a
commutator that distributes values in sequence to u, w and v .
2
1
Fractional delays, z − 3 and z − 3 are needed to synchronize the streams.
The output commutator takes values from the streams in sequence.
For clarity, we omit the fractional delays and regard each terminal, ◦, as
holding its value until needed. Initial commutator position has zero delay.
The commutator direction is against the direction of the z −1 delays.
DSP and Digital Filters (2016-9045)
Multirate: 11 – 12 / 14
Summary
11: Multirate Systems
• Multirate Systems
• Building blocks
• Resampling Cascades
• Noble Identities
• Noble Identities Proof
• Upsampled z-transform
• Downsampled z-transform
• Downsampled Spectrum
• Power Spectral Density
• Perfect Reconstruction
• Commutators
• Summary
• MATLAB routines
• Multirate Building Blocks
1:K
◦ Upsample: X(z) → X(z K )
Invertible, Inserts K − 1 zeros between samples
DSP and Digital Filters (2016-9045)
Shrinks and replicates spectrum
Follow by LP filter to remove images
Multirate: 11 – 13 / 14
Summary
11: Multirate Systems
• Multirate Systems
• Building blocks
• Resampling Cascades
• Noble Identities
• Noble Identities Proof
• Upsampled z-transform
• Downsampled z-transform
• Downsampled Spectrum
• Power Spectral Density
• Perfect Reconstruction
• Commutators
• Summary
• MATLAB routines
• Multirate Building Blocks
1:K
◦ Upsample: X(z) → X(z K )
Invertible, Inserts K − 1 zeros between samples
DSP and Digital Filters (2016-9045)
Shrinks and replicates spectrum
Follow by LP filter to remove images
◦ Downsample: X(z)
K:1 1
→ K
PK−1
−j2πk
K
1
K
z )
Destroys information and energy, keeps every K th sample
k=0
X(e
Expands and aliasses the spectrum
Spectrum is the average of K aliased expanded versions
Precede by LP filter to prevent aliases
Multirate: 11 – 13 / 14
Summary
11: Multirate Systems
• Multirate Systems
• Building blocks
• Resampling Cascades
• Noble Identities
• Noble Identities Proof
• Upsampled z-transform
• Downsampled z-transform
• Downsampled Spectrum
• Power Spectral Density
• Perfect Reconstruction
• Commutators
• Summary
• MATLAB routines
• Multirate Building Blocks
1:K
◦ Upsample: X(z) → X(z K )
Invertible, Inserts K − 1 zeros between samples
Shrinks and replicates spectrum
Follow by LP filter to remove images
◦ Downsample: X(z)
K:1 1
→ K
PK−1
−j2πk
K
1
K
z )
Destroys information and energy, keeps every K th sample
k=0
X(e
Expands and aliasses the spectrum
Spectrum is the average of K aliased expanded versions
Precede by LP filter to prevent aliases
• Equivalences
◦ Noble Identities: H(z) ←→ H(z K )
◦ Interchange P : 1 and 1 : Q iff P and Q coprime
DSP and Digital Filters (2016-9045)
Multirate: 11 – 13 / 14
Summary
11: Multirate Systems
• Multirate Systems
• Building blocks
• Resampling Cascades
• Noble Identities
• Noble Identities Proof
• Upsampled z-transform
• Downsampled z-transform
• Downsampled Spectrum
• Power Spectral Density
• Perfect Reconstruction
• Commutators
• Summary
• MATLAB routines
• Multirate Building Blocks
1:K
◦ Upsample: X(z) → X(z K )
Invertible, Inserts K − 1 zeros between samples
Shrinks and replicates spectrum
Follow by LP filter to remove images
◦ Downsample: X(z)
K:1 1
→ K
PK−1
−j2πk
K
1
K
z )
Destroys information and energy, keeps every K th sample
k=0
X(e
Expands and aliasses the spectrum
Spectrum is the average of K aliased expanded versions
Precede by LP filter to prevent aliases
• Equivalences
◦ Noble Identities: H(z) ←→ H(z K )
◦ Interchange P : 1 and 1 : Q iff P and Q coprime
• Commutators
◦ Combine delays and down/up sampling
DSP and Digital Filters (2016-9045)
Multirate: 11 – 13 / 14
Summary
11: Multirate Systems
• Multirate Systems
• Building blocks
• Resampling Cascades
• Noble Identities
• Noble Identities Proof
• Upsampled z-transform
• Downsampled z-transform
• Downsampled Spectrum
• Power Spectral Density
• Perfect Reconstruction
• Commutators
• Summary
• MATLAB routines
• Multirate Building Blocks
1:K
◦ Upsample: X(z) → X(z K )
Invertible, Inserts K − 1 zeros between samples
Shrinks and replicates spectrum
Follow by LP filter to remove images
◦ Downsample: X(z)
K:1 1
→ K
PK−1
−j2πk
K
1
K
z )
Destroys information and energy, keeps every K th sample
k=0
X(e
Expands and aliasses the spectrum
Spectrum is the average of K aliased expanded versions
Precede by LP filter to prevent aliases
• Equivalences
◦ Noble Identities: H(z) ←→ H(z K )
◦ Interchange P : 1 and 1 : Q iff P and Q coprime
• Commutators
◦ Combine delays and down/up sampling
For further details see Mitra: 13.
DSP and Digital Filters (2016-9045)
Multirate: 11 – 13 / 14
MATLAB routines
11: Multirate Systems
• Multirate Systems
• Building blocks
• Resampling Cascades
• Noble Identities
• Noble Identities Proof
• Upsampled z-transform
• Downsampled z-transform
• Downsampled Spectrum
• Power Spectral Density
• Perfect Reconstruction
• Commutators
• Summary
• MATLAB routines
resample
DSP and Digital Filters (2016-9045)
change sampling rate
Multirate: 11 – 14 / 14