DSP-CIS
Part-I / Chapter-2 : Signals & Systems Review
Marc Moonen & Toon van Waterschoot
Dept. E.E./ESAT-STADIUS, KU Leuven
[email protected]
www.esat.kuleuven.be/stadius/
Chapter-2 : Signals & Systems Review
• Discrete-Time/Digital Signals (10 slides)
Sampling, quantization, reconstruction
• Discrete-Time Systems (13 slides)
LTI, impulse response, convolution, z-transform, frequency
response, frequency spectrum, IIR/FIR
• Discrete Fourier Transform (4 slides)
DFT-IDFT, FFT
• Multi-Rate Systems
(11 slides)
DSP-CIS 2015 / Part-I / Chapter-2: Signals & Systems Review
2 / 40
Discrete-Time/Digital Signals
1/10
Jean-Baptiste Joseph Fourier (1768-1830)
Analog signal processing
Analog domain
(continuous-time domain)
Analog
Signal
Processing
Circuit
Analog IN
u (t )
+¥
U( f ) = F{u(t)} = ò u(t).e
- j 2 p . f .t
dt
Analog OUT
y (t )
Y( f ) = F{y(t)} =
-¥
+¥
ò y(t).e
- j 2 p . f .t
dt
-¥
(Fourier Transform / spectrum, where f = frequency)
DSP-CIS 2015 / Part-I / Chapter-2: Signals & Systems Review
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Discrete-Time/Digital Signals
2/10
Digital signal processing
Analog
domain
Analog IN
Digital
domain
Analog-toDigital
Conversion
011010
0101
100110
0010
DSP
Digital
IN
sampling
& quantization
DSP-CIS 2015 / Part-I / Chapter-2: Signals & Systems Review
Digital
OUT
Analog
domain
Digital-toAnalog
Conversion
Analog OUT
reconstruction
4 / 40
Discrete-Time/Digital Signals
1. Sampling
x[k ]  x(k .Ts )
x(t )
amplitude
3/10
amplitude
continuous-time
signal
discrete-time
signal
impulse
train
Ts
continuous-time (t)
01234
It will turn out (p.24-25) that a spectrum can be
computed from x[k] (=discrete-time), which
(remarkably) will be equal to the spectrum
(=Fourier transform) of the (continuous-time)
sequence of impulses…...
DSP-CIS 2015 / Part-I / Chapter-2: Signals & Systems Review
discrete-time [k]

xD (t )  x(t ).   (t  k .Ts )
k  
5 / 40
Discrete-Time/Digital Signals
4/10
So what does this spectrum of xD(t) look like…
• Spectrum replication
– Time domain:
+¥
xD (t) = x(t). å d (t - k.Ts )
k=-¥
– Frequency domain:
X( f ) = F{x(t)}
1 +¥
k
XD ( f ) = . å X( f - )
Ts k=-¥
Ts
magnitude
frequency (f)
DSP-CIS 2015 / Part-I / Chapter-2: Signals & Systems Review
XD ( f ) = F{xD (t)}
magnitude
frequency (f)
6 / 40
Discrete-Time/Digital Signals
5/10
• Sampling theorem
– Analog signal spectrum X(f) has a bandwidth of fmax Hz
– Spectrum replicas are separated by fs =1/Ts Hz
magnitude
frequency
– No spectral overlap if and only if
DSP-CIS 2015 / Part-I / Chapter-2: Signals & Systems Review
fs >
2.fmax
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Discrete-Time/Digital Signals
6/10
• Sampling theorem
– Analog signal spectrum X(f) has a bandwidth of fmax Hz
– Spectrum replicas are separated by fs =1/Ts Hz
magnitude
frequency
– Spectral overlap (=‘folding’, ‘aliasing’) if
DSP-CIS 2015 / Part-I / Chapter-2: Signals & Systems Review
fs <
2.fmax
8 / 40
Discrete-Time/Digital Signals
(*)
– Terminology:
• sampling frequency/rate fs
• Nyquist frequency fs/2
• sampling interval/period Ts
– E.g. CD audio: fs = 44,1 kHz
Harry Nyquist (1889 –1976)
• Sampling theorem
7/10
• Anti-aliasing prefilters
– If
then frequencies above the Nyquist
frequency are ‘folded’ into lower frequencies
(=aliasing)
– To avoid aliasing, sampling is usually preceded by
(analog-domain) low-pass (=anti-aliasing) filtering
(*) An equivalent formulation is fs > fmax-(-fmax) = fmax-fmin = ‘bandwidth’…will use this in p.36
DSP-CIS 2015 / Part-I / Chapter-2: Signals & Systems Review
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Discrete-Time/Digital Signals
2. B-bit quantization
discrete-time signal
amplitude
6dB per bit rule:
Number of bits B = log2 (
8/10
quantized discrete-time signal
=discrete-amplitude&time signal
=digital signal
amplitude
x[k ]
xQ [k ]
3Q
2Q
Q
0
-Q
-2Q
-3Q
discrete time [k]
R
discrete time [k]
range R
+1)
quantization step Q
Signal-to-QuantizationNoise-Ratio = 20 log10 (
Ex: CD audio = 16bits ~ 96dB SNR
range R
) = 6B dB
(LP’s: 60dB SNR)
quantization step Q
DSP-CIS 2015 / Part-I / Chapter-2: Signals & Systems Review
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Discrete-Time/Digital Signals
9/10
3. Reconstruction
– Reconstruction = ‘fill the gaps’ between adjacent samples
– Example: staircase reconstructor
amplitude
x[k ]
discretetime/digital signal
discrete time [k]
amplitude xR (t )
reconstructed
analog signal
continuous time (t)
– In a practical realization xD(t) is generated first as an intermediate
signal by means of a D-to-A & sampler, which is then followed by
(analog domain) filtering (details omitted)
DSP-CIS 2015 / Part-I / Chapter-2: Signals & Systems Review
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Discrete-Time/Digital Signals
10/10
• Complete scheme is…
Analog IN
Analog OUT
antialiasing
prefilter
sampler
antiimage
postfilter
quantizer
DSP
Digital
IN
DSP-CIS 2015 / Part-I / Chapter-2: Signals & Systems Review
reconstructor
Digital
OUT
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Discrete-Time Systems 1/13
Discrete-time system is `sampled data’ system
u[k]
y[k]
Input signal u[k] is a sequence of samples (=numbers)
..,u[-2],u[-1] ,u[0], u[1],u[2],…
System then produces a sequence of output samples y[k]
..,y[-2],y[-1] ,y[0], y[1],y[2],…
Example: `DSP’ block in previous slide
DSP-CIS 2015 / Part-I / Chapter-2: Signals & Systems Review
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Discrete-Time Systems 2/13
Will consider linear time-invariant (LTI) systems
u[k]
y[k]
Linear :
input u1[k] -> output y1[k]
input u2[k] -> output y2[k]
hence a.u1[k]+b.u2[k]-> a.y1[k]+b.y2[k]
Time-invariant (shift-invariant)
input u[k] -> output y[k]
hence input u[k-T] -> output y[k-T]
DSP-CIS 2015 / Part-I / Chapter-2: Signals & Systems Review
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Discrete-Time Systems 3/13
Causal systems
iff for all input signals with u[k]=0,k<0 -> output y[k]=0,k<0
Impulse response
K=0
input …,0,0, 1 ,0,0,0,...-> output …,0,0, h[0] ,h[1],h[2],h[3],...
General input u[0],u[1],u[2],u[3]
(cfr. linearity & shift-invariance!)
0
0
0 
 y[0] h[0]
 y[1]   h[1] h[0]
 u[0]
0
0


 

 y[2] h[2] h[1] h[0]
0   u[1] 


.
h[2] h[1] h[0] u[2]
 y[3]  0


 y[4]  0
0
h[2] h[1]  u[3]

 

0
0
h[2]
 y[5]  0
DSP-CIS 2015 / Part-I / Chapter-2: Signals & Systems Review
Otto Toeplitz (1881–1940)
K=0
this is called a
`Toeplitz’ matrix
15 / 40
Discrete-Time Systems 4/13
Convolution
u[0],u[1],u[2],u[3]
0
0 
 y[0] h[0] 0
 y[1]   h[1] h[0] 0
 u[0]
0


 

 y[2] h[2] h[1] h[0] 0   u[1] 



.
 y[3]  0 h[2] h[1] h[0] u[2]


 y[4]  0

0 h[2] h[1] u[3]

 

0
0 h[2]
 y[5]  0
D
y[k] = å h[k - k ].u[k ]= h[k]* u[k]
k
DSP-CIS 2015 / Part-I / Chapter-2: Signals & Systems Review
y[0],y[1],...
h[0],h[1],h[2],0,0,...
= `convolution sum‘
(=more convenient than Toeplitz matrix notation
when considering (infinitely) long input and impulse
response sequences
16 / 40
Discrete-Time Systems 5/13
Z-Transform of system h[k] and signals u[k],y[k]
Definition:
D
D
H (z)= å h[k].z
U(z)= å u[k].z
k
k
-k
-k
D
Y(z)= å y[k].z-k
k
Input/output relation:
0
0
0 
 y[0]
 h[0]
 y[1] 
 h[1] h[0]
 u[0]
0
0





 y[ 2]
h[2] h[1] h[0]
0   u[1] 
1
2
3
4
5
1
2
3
4
5
1 z
z
z
z
z .
z
z
z
z .
 1 z
.
y
[
3
]
0
h
[
2
]
h
[
1
]
h
[
0
]



 u[ 2]


 y[ 4]
 0
0
h[ 2] h[1]  u[3]




0
0
h[ 2]
 y[5]
 0
 


Y ( z)


H ( z).1 z1
 Y ( z )  H ( z ).U ( z )
DSP-CIS 2015 / Part-I / Chapter-2: Signals & Systems Review





z2
z3





H(z) is `transfer function’
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Discrete-Time Systems 6/13
Z-Transform
• Easy input-output relation: Y ( z )  H ( z ).U ( z )
• May be viewed as `shorthand’ notation
(for convolution operation/Toeplitz-vector product)
• Stability
=bounded input u[k] leads to bounded output y[k]
--iff
 h[k ]  
k
--iff poles of H(z) inside the unit circle
(now z=complex variable)
(for causal,rational systems, see below)
DSP-CIS 2015 / Part-I / Chapter-2: Signals & Systems Review
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Discrete-Time Systems 7/13
Example-1 : `Delay operator’
K=0
u[k]
Impulse response is …,0,0, 0 ,1,0,0,0,…
Transfer function is
H (z) = z-1 =
Pole at z=0

1
z
Example-2 : Delay + feedback
K=0
Impulse response is …,0,0, 0 ,1,a,a^2,a^3…
Transfer function is H ( z )  z 1  a.z 2  a 2 .z 3  a 3 .z 4  ...
Pole at z=a
y[k]=u[k-1]
1
 H ( z )  a.z H ( z )  z
1
z 1
1
 H ( z) 

1  a.z 1 z  a
DSP-CIS 2015 / Part-I / Chapter-2: Signals & Systems Review
u[k]
+
y[k]

x
a
=simple rational function
realized with a delay element,
a multiplier and an adder
19 / 40
Discrete-Time Systems 8/13
Will consider only rational transfer functions:
B(z) b0 zL + b1zL-1 +... + bL b0 + b1z-1 +... + bL z- L
H (z) =
= L
=
L-1
A(z)
z + a1z +... + aL
1+ a1z-1 +... + aL z-L
• L poles (zeros of A(z)) , L zeros (zeros of B(z))
• Corresponds to difference equation
Y ( z )  H ( z ).U ( z )  A( z ).Y ( z )  B( z ).U ( z )  ...
y[k] = b0.u[k]+ b1.u[k -1]+... + bL .u[k- L]- a1.y[k -1]-... - aL.y[k - L]
• Hence rational H(z) can be realized with finite number of delay
elements, multipliers and adders
• In general, this is a `infinitely long impulse response’ (`IIR’) system
(as in example-2)
DSP-CIS 2015 / Part-I / Chapter-2: Signals & Systems Review
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Discrete-Time Systems 9/13
Special case is
B(z)
-1
-L
H (z) = L = b0 + b1z +... + bL z
z
• L poles at the origin z=0 (hence guaranteed stability)
• L zeros (zeros of B(z))
= `all zero’ filter
• Corresponds to difference equation
Y(z) = H(z).U(z) Þ y[k] = b0.u[k]+ b1.u[k -1]+... + bL .u[k - L]
=`moving average’ (MA) filter
• Impulse response h[k] is
0, 0, 0, b0 , b1,..., bL-1, bL , 0, 0, 0,...
= `finite impulse response’ (`FIR’) filter
DSP-CIS 2015 / Part-I / Chapter-2: Signals & Systems Review
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Discrete-Time Systems 10/13
H(z) & frequency response:
• Given a system H(z)
• Given an input signal = complex exponential
u[k ]  e
jk
Im
u[2]
  k  
u[1]

Re
 cos(k )  j. sin( k )
u[0]=1
(where ω=radial frequency)
• Output signal :
y[k] = å h[k ].u[k-k ] = å h[k ].ejw (k-k )=e jwk å h[k ].e- jwk = u[k].H(ejw )
k
k
j
H (e )
k
= `frequency response’
= complex function of radial frequency ω
= H(z) evaluated on the unit circle
DSP-CIS 2015 / Part-I / Chapter-2: Signals & Systems Review
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Discrete-Time Systems 11/13
H(z) & frequency response:
• Periodic with period = 2
j
H (e )
• For a real-valued impulse response h[k]
- magnitude response H (e j ) is even function
- phase response
is odd function
j
H (e )
• Example-1: Low-pass filter
1
0.5
Nyquist frequency
e
jk
 ...,1,1,1,1,1,...
0
-4
5

-2
0
2
-2
0
2

4

0
(=2 samples/period)
-5
-4
• Example-2: All-pass filter H (e j )  1
DSP-CIS 2015 / Part-I / Chapter-2: Signals & Systems Review
4
DC
ej 0k =...,1,1,1,1,1,...
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Discrete-Time Systems 12/13
• Z-Transform & Discrete-Time Fourier Transform
z = ejw
Y(z) = H(z).U(z)
Y (e j )  H (e j ).U (e j )
– H (e j ) is frequency response of the LTI system
– U(ejw ) is frequency spectrum (‘Discrete-Time Fourier Transform’)
of input signal
U(ejw ) = U(z) z=ejw =
+¥
-k
u[k].z
å
k=-¥
=
z=ejw
+¥
- jw .k
u[k].
e
å
k=-¥
(compare to Fourier Transform, see p.3)
jw
– Y(e ) is frequency spectrum of the output signal
DSP-CIS 2015 / Part-I / Chapter-2: Signals & Systems Review
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Discrete-Time Systems 13/13
• Z-Transform & Fourier Transform
It is proved that…
j
– The frequency response H (e ) of an LTI system is equal to the
Fourier transform of the continuous-time impulse
f
e.g. f = s Þ w = p
2
sequence (see p.5) constructed with h[k]

H (e ) = ... = F{hD (t)} = F{å h[k].d (t - k.Ts )} , w = 2p .
jw
k
jw
f
fs
jw
– The frequency spectrum U(e ) or Y(e ) of a discrete-time signal
is equal to the Fourier transform of the continuous-time impulse
sequence constructed with u[k] or y[k]
f
U(e ) =... = F{uD (t)} = F{åu[k].d (t - k.Ts )} , w = 2p .
fs
k
jw
– F{yD (t)} = F{hD (t)}.F{uD (t)} corresponds to continuous-time
Y( f ) = H( f ).U( f ) iff U( f ),Y( f ), H( f ) are bandlimited (no aliasing)
DSP-CIS 2015 / Part-I / Chapter-2: Signals & Systems Review
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Discrete/Fast Fourier Transform 1/4
• DFT definition:
– The `Discrete-time Fourier Transform’ of a discrete-time
system/signal x[k] is a (periodic) continuous function of
the radial frequency ω (see p.28)
X (e
j
 

)    x[k ]. z  k 
 k  
 z  e j
– The `Discrete Fourier Transform’ (DFT) is a discretized
version of this, obtained by sampling ω at N uniformly
spaced frequencies wn = 2p .n / N
(n=0,1,..,N-1)
and by truncating x[k] to N samples (k=0,1,..,N-1)
j
X[e
2 p .n
N
N-1
-j
] = å x[k]. e
2 p .n
k
N
k=0
DSP-CIS 2015 / Part-I / Chapter-2: Signals & Systems Review
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Discrete/Fast Fourier Transform 2/4
• DFT & Inverse DFT (IDFT):
– An N-point DFT sequence can be calculated from an
N-point time sequence:
j
X[e
2 p .n
N
N-1
-j
] = å x[k]. e
2 p .n
k
N
= DFT
k=0
– Conversely, an N-point time sequence can be calculated
from an N-point DFT sequence:
j
1 N-1
x[k] =
X[e
å
N n=0
2p .n
N
DSP-CIS 2015 / Part-I / Chapter-2: Signals & Systems Review
j
]. e
2 p .n
k
N
= IDFT
27 / 40
Discrete/Fast Fourier Transform 3/4
• DFT/IDFT in matrix form
– Using shorthand notation..
– ..the DFT can be rewritten as
X = F.x
– ..the IDFT can be rewritten as
x = F-1.X
1
= F *.X
N
DSP-CIS 2015 / Part-I / Chapter-2: Signals & Systems Review
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Discrete/Fast Fourier Transform 4/4
• Fast Fourier Transform (FFT) (1805/1965)
Split up N-point DFT in two N/2-point DFT’s
Split up two N/2-point DFT’s in four N/4-point DFT’s
…
Split up N/2 2-point DFT’s in N 1-point DFT’s
Calculate N 1-point DFT’s
Rebuild N/2 2-point DFT’s from N 1-point DFT’s
…
Rebuild two N/2-point DFT’s from four N/4-point DFT’s
Rebuild N-point DFT from two N/2-point DFT’s
– DFT complexity of N2 multiplications is reduced to
FFT complexity of O(N.log2(N)) multiplications
James W. Cooley
•
•
•
•
•
•
•
•
•
John W.Tukey
Carl Friedrich Gauss (1777-1855)
– Divide-and-conquer approach:
– Similar IFFT
DSP-CIS 2015 / Part-I / Chapter-2: Signals & Systems Review
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Multi-Rate Systems 1/11
• Decimation : decimator (=downsampler)
u[0],u[1],u[2]...
D
u[0],
u[N],
u[2D]...
Example : u[k]: 1,2,3,4,5,6,7,8,9,…
2-fold downsampling: 1,3,5,7,9,...
• Interpolation : expander (=upsampler)
u[0],
u[1],
u[2],...
D
u[0],0,..0,u[1],0,…,0,u[2]...
Example : u[k]: 1,2,3,4,5,6,7,8,9,…
2-fold upsampling: 1,0,2,0,3,0,4,0,5,0...
DSP-CIS 2015 / Part-I / Chapter-2: Signals & Systems Review
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Multi-Rate Systems 2/11
• Z-transform & frequency domain analysis of expander
u[0],
u[1],
u[2],...
U (z )
D
u[0],0,..0,u[1],0,…,0,u[2]...
D
U(zD )
`images’
z  e j
3




 xHz
 xHz
`Expansion in time domain ~ compression in frequency domain’
DSP-CIS 2015 / Part-I / Chapter-2: Signals & Systems Review
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Multi-Rate Systems 3/11
• Z-transform & frequency domain analysis of expander
u[0],
u[1],
u[2],...
D
u[0],0,..0,u[1],0,…,0,u[2]...
Expander mostly followed by `interpolation filter’ to remove images
(and `interpolate the zeros’)
`images’
3


LP


 xHz
 xHz


 xHz
Interpolation filter can be low-/band-/high-pass (see p.35-36 and Chapter-10)
DSP-CIS 2015 / Part-I / Chapter-2: Signals & Systems Review
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Multi-Rate Systems 4/11
• Z-transform & frequency domain analysis of decimator
u[0],u[1],u[2]...
U (z )
D
u[0],
u[D],
u[2D]...
1 D-1
.åU(z1 D .e- j 2 p d
D d=0
D
d=2
D
)
d=0
d=1
3


3

-3p -2p
 xHz


2p
3p
 xHz
`Compression in time domain ~ expansion in frequency domain’
jw /D
) is periodic with period 2Dp
PS: Note that U (e j ) is periodic with period 2 while U(e
The summation with d=0…D-1 restores the periodicity with period 2 !
DSP-CIS 2015 / Part-I / Chapter-2: Signals & Systems Review
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Multi-Rate Systems 5/11
• Z-transform & frequency domain analysis of decimator
u[0],u[1],u[2]...
D
u[0],
u[N],
u[2D]...
Decimation introduces ALIASING if input signal occupies frequency band
larger than 2p D , hence mostly preceded by anti-aliasing (decimation)
filter
d=2
d=0
d=1
3
LP


3

 xHz


 xHz
Anti-aliasing filter can be low-/band-/high-pass (see p.35-36 and Chapter-10)
DSP-CIS 2015 / Part-I / Chapter-2: Signals & Systems Review
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Multi-Rate Systems 6/11
• Example: LP anti-aliasing / down / up / LP interpolation
fmax
LP


3

(*)


3

3


LP
3
Will be used in Part IV on ‘Filterbanks’

DSP-CIS
2015 / Part-I /toChapter-2:
& Systems
Review
(*) Corresponds
NyquistSignals
theorem:
3-fold
reduction fmax 

3-fold reduction fs

3

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Multi-Rate Systems 7/11
• Example: HP anti-aliasing / down / up / HP interpolation
fmin fmax
HP


3

(*)


3

3


HP
3

Will be used in Part IV on ‘Filterbanks’
3p
3



3

DSP-CIS
2015 / Part-I
/ Chapter-2:
Signals
Systems Review
/ 40 )
(*) Corresponds
to Nyquist
theorem
for &‘passband’
signals: fs > fmax-fmin (as in footnote p.9, now fmin ≠36
-fmax
Multi-Rate Systems 8/10
• Interconnection of multi-rate building blocks
D
x
D
=
x
a
u1[k]
a
D
+
=
u2[k]
u1[k]
D
x
u2[k]
=
u1[k]
D
u2[k]
D
u1[k]
D
u2[k]
D
+
x
i.e. all filter operations can be performed at the lowest rate!
Identities also hold if decimators are replaced by expanders
DSP-CIS 2015 / Part-I / Chapter-2: Signals & Systems Review
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Multi-Rate Systems 9/11
• `Noble identities‘(only for rational functions)
u[k]
y[k]
D
H(z )
D
u[k]
u[k]
H (z )
D
D
=
y[k]
y[k]
H (z )
u[k]
=
DSP-CIS 2015 / Part-I / Chapter-2: Signals & Systems Review
y[k]
D
D
H(z )
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Multi-Rate Systems 10/10
Application of `noble identities : efficient multi-rate realizations of FIR filters through…
• Polyphase decomposition:
Example : (2-fold decomposition)
H ( z )  h[0]  h[1]. z 1  h[2]. z 2  h[3]. z 3  h[4]. z 4  h[5]. z 5  h[6]. z 6
 (h[0]  h[2]. z  2  h[4]. z  4  h[6]. z 6 )  z 1.(h[1]  h[3]. z  2  h[5]. z  4 )



E0 ( z 2 )
E1 ( z 2 )
Example : (3-fold decomposition)
H ( z )  h[0]  h[1]. z 1  h[ 2]. z 2  h[3]. z 3  h[4]. z 4  h[5]. z 5  h[6]. z 6
 (h[0]  h[3]. z 3  h[6]. z 6 )  z 1.( h[1]  h[ 4]. z 3 )  z  2 .( h[ 2]  h[5]. z 3 )





E0 ( z 3 )
E1 ( z 3 )
E2 ( z 3 )
General: (D-fold decomposition)
H (z) =
¥
å h[k].z
k=-¥
-k
D-1
= å z .Ed (z )
-d
D
d=0
DSP-CIS 2015 / Part-I / Chapter-2: Signals & Systems Review
,
Ed (z) =
¥
å h[D.k + d].z
-k
k=-¥
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Multi-Rate Systems 11/11
• Polyphase decomposition:
Example : efficient realization of FIR decimation/interpolation filter
u[k]
H(z)
=
2
E0 ( z )
z 1

2
E1 ( z )
u[k]
2
E1 ( z 2 )
E0 ( z 2 )
+
H(z)

z 1
u[k]
z 1
2
=
+
2
E0 ( z )
2
E1 ( z )

u[k]
E1 ( z )
2
E0 ( z )
2
+

z 1
+
i.e. all filter operations can be performed at the lowest rate!
DSP-CIS 2015 / Part-I / Chapter-2: Signals & Systems Review
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