2: Three Different Fourier
Transforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Summary
• MATLAB routines
DSP and Digital Filters (2016-8746)
2: Three Different Fourier
Transforms
Fourier Transforms: 2 – 1 / 13
Fourier Transforms
2: Three Different Fourier
Transforms
Three different Fourier Transforms:
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Summary
• MATLAB routines
DSP and Digital Filters (2016-8746)
Fourier Transforms: 2 – 2 / 13
Fourier Transforms
2: Three Different Fourier
Transforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Summary
• MATLAB routines
Three different Fourier Transforms:
• CTFT (Continuous-Time Fourier Transform): x(t) → X(jΩ)
DSP and Digital Filters (2016-8746)
Fourier Transforms: 2 – 2 / 13
Fourier Transforms
2: Three Different Fourier
Transforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Summary
• MATLAB routines
Three different Fourier Transforms:
• CTFT (Continuous-Time Fourier Transform): x(t) → X(jΩ)
• DTFT (Discrete-Time Fourier Transform): x[n] → X(ejω )
DSP and Digital Filters (2016-8746)
Fourier Transforms: 2 – 2 / 13
Fourier Transforms
2: Three Different Fourier
Transforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Summary
• MATLAB routines
Three different Fourier Transforms:
• CTFT (Continuous-Time Fourier Transform): x(t) → X(jΩ)
• DTFT (Discrete-Time Fourier Transform): x[n] → X(ejω )
• DFT a.k.a. FFT (Discrete Fourier Transform): x[n] → X[k]
DSP and Digital Filters (2016-8746)
Fourier Transforms: 2 – 2 / 13
Fourier Transforms
2: Three Different Fourier
Transforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Summary
• MATLAB routines
Three different Fourier Transforms:
• CTFT (Continuous-Time Fourier Transform): x(t) → X(jΩ)
• DTFT (Discrete-Time Fourier Transform): x[n] → X(ejω )
• DFT a.k.a. FFT (Discrete Fourier Transform): x[n] → X[k]
Forward Transform
CTFT
DSP and Digital Filters (2016-8746)
X(jΩ) =
R∞
x(t)e
−∞
−jΩt
Inverse Transform
dt
x(t) =
1
2π
R∞
jΩt
X(jΩ)e
dΩ
−∞
Fourier Transforms: 2 – 2 / 13
Fourier Transforms
2: Three Different Fourier
Transforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Summary
• MATLAB routines
Three different Fourier Transforms:
• CTFT (Continuous-Time Fourier Transform): x(t) → X(jΩ)
• DTFT (Discrete-Time Fourier Transform): x[n] → X(ejω )
• DFT a.k.a. FFT (Discrete Fourier Transform): x[n] → X[k]
Forward Transform
CTFT
DTFT
DSP and Digital Filters (2016-8746)
R∞
dt
X(jΩ) = −∞ x(t)e
P∞
jω
X(e ) = −∞ x[n]e−jωn
−jΩt
Inverse Transform
x(t) =
x[n] =
1
2π
1
2π
R∞
jΩt
X(jΩ)e
dΩ
−∞
Rπ
jω jωn
X(e
)e dω
−π
Fourier Transforms: 2 – 2 / 13
Fourier Transforms
2: Three Different Fourier
Transforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Summary
• MATLAB routines
Three different Fourier Transforms:
• CTFT (Continuous-Time Fourier Transform): x(t) → X(jΩ)
• DTFT (Discrete-Time Fourier Transform): x[n] → X(ejω )
• DFT a.k.a. FFT (Discrete Fourier Transform): x[n] → X[k]
Forward Transform
CTFT
DTFT
R∞
dt
X(jΩ) = −∞ x(t)e
P∞
jω
X(e ) = −∞ x[n]e−jωn
−jΩt
Inverse Transform
x(t) =
x[n] =
1
2π
1
2π
R∞
jΩt
X(jΩ)e
dΩ
−∞
Rπ
jω jωn
X(e
)e dω
−π
We use Ω for “real” and ω = ΩT for “normalized” angular frequency.
f
Nyquist frequency is at ΩNyq = 2π 2s = Tπ and ωNyq = π .
DSP and Digital Filters (2016-8746)
Fourier Transforms: 2 – 2 / 13
Fourier Transforms
2: Three Different Fourier
Transforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Summary
• MATLAB routines
Three different Fourier Transforms:
• CTFT (Continuous-Time Fourier Transform): x(t) → X(jΩ)
• DTFT (Discrete-Time Fourier Transform): x[n] → X(ejω )
• DFT a.k.a. FFT (Discrete Fourier Transform): x[n] → X[k]
Forward Transform
CTFT
DTFT
DFT
R∞
dt
X(jΩ) = −∞ x(t)e
P∞
jω
X(e ) = −∞ x[n]e−jωn
PN −1
kn
X[k] = 0
x[n]e−j2π N
−jΩt
Inverse Transform
x(t) =
x[n] =
x[n] =
1
2π
1
2π
1
N
R∞
jΩt
X(jΩ)e
dΩ
−∞
Rπ
jω jωn
X(e
)e dω
−π
PN −1
j2π kn
N
X[k]e
0
We use Ω for “real” and ω = ΩT for “normalized” angular frequency.
f
Nyquist frequency is at ΩNyq = 2π 2s = Tπ and ωNyq = π .
DSP and Digital Filters (2016-8746)
Fourier Transforms: 2 – 2 / 13
Fourier Transforms
2: Three Different Fourier
Transforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Summary
• MATLAB routines
Three different Fourier Transforms:
• CTFT (Continuous-Time Fourier Transform): x(t) → X(jΩ)
• DTFT (Discrete-Time Fourier Transform): x[n] → X(ejω )
• DFT a.k.a. FFT (Discrete Fourier Transform): x[n] → X[k]
Forward Transform
CTFT
DTFT
DFT
R∞
dt
X(jΩ) = −∞ x(t)e
P∞
jω
X(e ) = −∞ x[n]e−jωn
PN −1
kn
X[k] = 0
x[n]e−j2π N
−jΩt
Inverse Transform
x(t) =
x[n] =
x[n] =
1
2π
1
2π
1
N
R∞
jΩt
X(jΩ)e
dΩ
−∞
Rπ
jω jωn
X(e
)e dω
−π
PN −1
j2π kn
N
X[k]e
0
We use Ω for “real” and ω = ΩT for “normalized” angular frequency.
f
Nyquist frequency is at ΩNyq = 2π 2s = Tπ and ωNyq = π .
For “power signals” (energy ∝ time interval), CTFT & DTFT are unbounded.
Fix this by normalizing:
RA
1
X(jΩ) = limA→∞ 2A −A x(t)e−jΩt dt
PA
1
jω
X(e ) = limA→∞ 2A −A x[n]e−jωn
DSP and Digital Filters (2016-8746)
Fourier Transforms: 2 – 2 / 13
Convergence of DTFT
2: Three Different Fourier
Transforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Summary
• MATLAB routines
DTFT: X(ejω ) =
P∞
x[n]e−jωn does not converge for all x[n].
PK
jω
−jωn
Consider the finite sum: XK (e ) =
−K x[n]e
DSP and Digital Filters (2016-8746)
−∞
Fourier Transforms: 2 – 3 / 13
Convergence of DTFT
2: Three Different Fourier
Transforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Summary
• MATLAB routines
DTFT: X(ejω ) =
P∞
x[n]e−jωn does not converge for all x[n].
PK
jω
−jωn
Consider the finite sum: XK (e ) =
−K x[n]e
−∞
Strong Convergence:
x[n] absolutely summable ⇒ X(ejω ) converges uniformly
DSP and Digital Filters (2016-8746)
Fourier Transforms: 2 – 3 / 13
Convergence of DTFT
2: Three Different Fourier
Transforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Summary
• MATLAB routines
DTFT: X(ejω ) =
P∞
x[n]e−jωn does not converge for all x[n].
PK
jω
−jωn
Consider the finite sum: XK (e ) =
−K x[n]e
−∞
Strong Convergence:
x[n] absolutely summable ⇒X(ejω ) converges uniformly
P∞
DSP and Digital Filters (2016-8746)
−∞
|x[n]| < ∞ ⇒ supω X(ejω ) − XK (ejω ) −−−−→ 0
K→∞
Fourier Transforms: 2 – 3 / 13
Convergence of DTFT
2: Three Different Fourier
Transforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Summary
• MATLAB routines
DTFT: X(ejω ) =
P∞
x[n]e−jωn does not converge for all x[n].
PK
jω
−jωn
Consider the finite sum: XK (e ) =
−K x[n]e
−∞
Strong Convergence:
x[n] absolutely summable ⇒X(ejω ) converges uniformly
P∞
−∞
|x[n]| < ∞ ⇒ supω X(ejω ) − XK (ejω ) −−−−→ 0
K→∞
Weaker convergence:
x[n] finite energy ⇒ X(ejω ) converges in mean square
DSP and Digital Filters (2016-8746)
Fourier Transforms: 2 – 3 / 13
Convergence of DTFT
2: Three Different Fourier
Transforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Summary
• MATLAB routines
DTFT: X(ejω ) =
P∞
x[n]e−jωn does not converge for all x[n].
PK
jω
−jωn
Consider the finite sum: XK (e ) =
−K x[n]e
−∞
Strong Convergence:
x[n] absolutely summable ⇒X(ejω ) converges uniformly
P∞
−∞
|x[n]| < ∞ ⇒ supω X(ejω ) − XK (ejω ) −−−−→ 0
K→∞
Weaker convergence:
x[n] finite energy ⇒ X(ejω ) converges in mean square
P∞
−∞
DSP and Digital Filters (2016-8746)
2
|x[n]| < ∞ ⇒
1
2π
Rπ X(ejω ) − XK (ejω )2 dω −−−−→ 0
−π
K→∞
Fourier Transforms: 2 – 3 / 13
Convergence of DTFT
2: Three Different Fourier
Transforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Summary
• MATLAB routines
DTFT: X(ejω ) =
P∞
x[n]e−jωn does not converge for all x[n].
PK
jω
−jωn
Consider the finite sum: XK (e ) =
−K x[n]e
−∞
Strong Convergence:
x[n] absolutely summable ⇒X(ejω ) converges uniformly
P∞
−∞
|x[n]| < ∞ ⇒ supω X(ejω ) − XK (ejω ) −−−−→ 0
K→∞
Weaker convergence:
x[n] finite energy ⇒ X(ejω ) converges in mean square
P∞
−∞
2
|x[n]| < ∞ ⇒
1
2π
Rπ X(ejω ) − XK (ejω )2 dω −−−−→ 0
−π
K→∞
0.5πn
Example: x[n] = sin πn
K=5
1
K=20
1
0.6
0.6
0.6
0.4
0.2
0.2
0
0
-0.2
0
DSP and Digital Filters (2016-8746)
0.1
0.2
0.3
0.4
ω/2π (rad/sample)
0.5
K
0.4
jω
0.8
jω
0.8
K
K
jω
1
0.8
-0.2
0
K=50
0.4
0.2
0
0.1
0.2
0.3
0.4
ω/2π (rad/sample)
0.5
-0.2
0
0.1
0.2
0.3
0.4
ω/2π (rad/sample)
0.5
Fourier Transforms: 2 – 3 / 13
Convergence of DTFT
2: Three Different Fourier
Transforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Summary
• MATLAB routines
DTFT: X(ejω ) =
P∞
x[n]e−jωn does not converge for all x[n].
PK
jω
−jωn
Consider the finite sum: XK (e ) =
−K x[n]e
−∞
Strong Convergence:
x[n] absolutely summable ⇒X(ejω ) converges uniformly
P∞
−∞
|x[n]| < ∞ ⇒ supω X(ejω ) − XK (ejω ) −−−−→ 0
K→∞
Weaker convergence:
x[n] finite energy ⇒ X(ejω ) converges in mean square
P∞
−∞
2
|x[n]| < ∞ ⇒
1
2π
Rπ X(ejω ) − XK (ejω )2 dω −−−−→ 0
−π
K→∞
0.5πn
Example: x[n] = sin πn
K=5
1
K=20
1
0.6
0.6
0.6
0.4
0.2
0.2
0
0
-0.2
0
0.1
0.2
0.3
0.4
ω/2π (rad/sample)
0.5
K
0.4
jω
0.8
jω
0.8
K
K
jω
1
0.8
-0.2
0
K=50
0.4
0.2
0
0.1
0.2
0.3
0.4
ω/2π (rad/sample)
0.5
-0.2
0
0.1
0.2
0.3
0.4
ω/2π (rad/sample)
0.5
Gibbs phenomenon:
Converges at each ω as K → ∞ but peak error does not get smaller.
DSP and Digital Filters (2016-8746)
Fourier Transforms: 2 – 3 / 13
DTFT Properties
2: Three Different Fourier
Transforms
DTFT: X(ejω ) =
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Summary
• MATLAB routines
DSP and Digital Filters (2016-8746)
P∞
−∞
x[n]e−jωn
Fourier Transforms: 2 – 4 / 13
DTFT Properties
2: Three Different Fourier
Transforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Summary
• MATLAB routines
DTFT: X(ejω ) =
P∞
x[n]e−jωn
• DTFT is periodic in ω : X(ej(ω+2mπ) ) = X(ejω ) for integer m.
DSP and Digital Filters (2016-8746)
−∞
Fourier Transforms: 2 – 4 / 13
DTFT Properties
2: Three Different Fourier
Transforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Summary
• MATLAB routines
DTFT: X(ejω ) =
P∞
x[n]e−jωn
• DTFT is periodic in ω : X(ej(ω+2mπ) ) = X(ejω ) for integer m.
−∞
evaluated at the point ejω :
• DTFT is the z -Transform
P∞
X(z) = −∞ x[n]z −n
DSP and Digital Filters (2016-8746)
Fourier Transforms: 2 – 4 / 13
DTFT Properties
2: Three Different Fourier
Transforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Summary
• MATLAB routines
DTFT: X(ejω ) =
P∞
x[n]e−jωn
• DTFT is periodic in ω : X(ej(ω+2mπ) ) = X(ejω ) for integer m.
−∞
evaluated at the point ejω :
• DTFT is the z -Transform
P∞
X(z) = −∞ x[n]z −n
DTFT converges iff the ROC includes |z| = 1.
DSP and Digital Filters (2016-8746)
Fourier Transforms: 2 – 4 / 13
DTFT Properties
2: Three Different Fourier
Transforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Summary
• MATLAB routines
DTFT: X(ejω ) =
P∞
x[n]e−jωn
• DTFT is periodic in ω : X(ej(ω+2mπ) ) = X(ejω ) for integer m.
−∞
evaluated at the point ejω :
• DTFT is the z -Transform
P∞
X(z) = −∞ x[n]z −n
DTFT converges iff the ROC includes |z| = 1.
• DTFT is the same as the CTFT of a signal comprising impulses at
the sample times (Dirac δ functions) of appropriate heights:
P
xδ (t) =
x[n]δ(t − nT )
DSP and Digital Filters (2016-8746)
Fourier Transforms: 2 – 4 / 13
DTFT Properties
2: Three Different Fourier
Transforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Summary
• MATLAB routines
DTFT: X(ejω ) =
P∞
x[n]e−jωn
• DTFT is periodic in ω : X(ej(ω+2mπ) ) = X(ejω ) for integer m.
−∞
evaluated at the point ejω :
• DTFT is the z -Transform
P∞
X(z) = −∞ x[n]z −n
DTFT converges iff the ROC includes |z| = 1.
• DTFT is the same as the CTFT of a signal comprising impulses at
the sample times (Dirac δ functions) of appropriate heights:
P
xδ (t) =
x[n]δ(t − nT )
Equivalent to multiplying a continuous x(t) by an impulse train.
DSP and Digital Filters (2016-8746)
Fourier Transforms: 2 – 4 / 13
DTFT Properties
2: Three Different Fourier
Transforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Summary
• MATLAB routines
DTFT: X(ejω ) =
P∞
x[n]e−jωn
• DTFT is periodic in ω : X(ej(ω+2mπ) ) = X(ejω ) for integer m.
−∞
evaluated at the point ejω :
• DTFT is the z -Transform
P∞
X(z) = −∞ x[n]z −n
DTFT converges iff the ROC includes |z| = 1.
• DTFT is the same as the CTFT of a signal comprising impulses at
the sample times (Dirac δ functions) of appropriate heights:
P
P∞
xδ (t) =
x[n]δ(t − nT )= x(t) × −∞ δ(t − nT )
Equivalent to multiplying a continuous x(t) by an impulse train.
DSP and Digital Filters (2016-8746)
Fourier Transforms: 2 – 4 / 13
DTFT Properties
2: Three Different Fourier
Transforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Summary
• MATLAB routines
DTFT: X(ejω ) =
P∞
x[n]e−jωn
• DTFT is periodic in ω : X(ej(ω+2mπ) ) = X(ejω ) for integer m.
−∞
evaluated at the point ejω :
• DTFT is the z -Transform
P∞
X(z) = −∞ x[n]z −n
DTFT converges iff the ROC includes |z| = 1.
• DTFT is the same as the CTFT of a signal comprising impulses at
the sample times (Dirac δ functions) of appropriate heights:
P
P∞
xδ (t) =
x[n]δ(t − nT )= x(t) × −∞ δ(t − nT )
Equivalent to multiplying a continuous x(t) by an impulse train.
P∞
jω
−jωn
Proof: X(e ) =
−∞ x[n]e
DSP and Digital Filters (2016-8746)
Fourier Transforms: 2 – 4 / 13
DTFT Properties
2: Three Different Fourier
Transforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Summary
• MATLAB routines
DTFT: X(ejω ) =
P∞
x[n]e−jωn
• DTFT is periodic in ω : X(ej(ω+2mπ) ) = X(ejω ) for integer m.
−∞
evaluated at the point ejω :
• DTFT is the z -Transform
P∞
X(z) = −∞ x[n]z −n
DTFT converges iff the ROC includes |z| = 1.
• DTFT is the same as the CTFT of a signal comprising impulses at
the sample times (Dirac δ functions) of appropriate heights:
P
P∞
xδ (t) =
x[n]δ(t − nT )= x(t) × −∞ δ(t − nT )
Equivalent to multiplying a continuous x(t) by an impulse train.
P∞
jω
−jωn
Proof: X(e ) =
−∞ x[n]eR
P∞
∞
−jω Tt
δ(t
−
nT
)e
x[n]
dt
n=−∞
−∞
DSP and Digital Filters (2016-8746)
Fourier Transforms: 2 – 4 / 13
DTFT Properties
2: Three Different Fourier
Transforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Summary
• MATLAB routines
DTFT: X(ejω ) =
P∞
x[n]e−jωn
• DTFT is periodic in ω : X(ej(ω+2mπ) ) = X(ejω ) for integer m.
−∞
evaluated at the point ejω :
• DTFT is the z -Transform
P∞
X(z) = −∞ x[n]z −n
DTFT converges iff the ROC includes |z| = 1.
• DTFT is the same as the CTFT of a signal comprising impulses at
the sample times (Dirac δ functions) of appropriate heights:
P
P∞
xδ (t) =
x[n]δ(t − nT )= x(t) × −∞ δ(t − nT )
Equivalent to multiplying a continuous x(t) by an impulse train.
P∞
jω
−jωn
Proof: X(e ) =
−∞ x[n]eR
P∞
∞
−jω Tt
δ(t
−
nT
)e
x[n]
dt
n=−∞
−∞
(i) R ∞ P∞
t
= −∞ n=−∞ x[n]δ(t − nT )e−jω T dt
(i) OK if
DSP and Digital Filters (2016-8746)
P∞
−∞
|x[n]| < ∞.
Fourier Transforms: 2 – 4 / 13
DTFT Properties
2: Three Different Fourier
Transforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Summary
• MATLAB routines
DTFT: X(ejω ) =
P∞
x[n]e−jωn
• DTFT is periodic in ω : X(ej(ω+2mπ) ) = X(ejω ) for integer m.
−∞
evaluated at the point ejω :
• DTFT is the z -Transform
P∞
X(z) = −∞ x[n]z −n
DTFT converges iff the ROC includes |z| = 1.
• DTFT is the same as the CTFT of a signal comprising impulses at
the sample times (Dirac δ functions) of appropriate heights:
P
P∞
xδ (t) =
x[n]δ(t − nT )= x(t) × −∞ δ(t − nT )
Equivalent to multiplying a continuous x(t) by an impulse train.
P∞
jω
−jωn
Proof: X(e ) =
−∞ x[n]eR
P∞
∞
−jω Tt
δ(t
−
nT
)e
x[n]
dt
n=−∞
−∞
(i) R ∞ P∞
t
= −∞ n=−∞ x[n]δ(t − nT )e−jω T dt
(ii) R ∞
= −∞ xδ (t)e−jΩt dt
P∞
(i) OK if
(ii) use ω = ΩT .
−∞ |x[n]| < ∞.
DSP and Digital Filters (2016-8746)
Fourier Transforms: 2 – 4 / 13
DFT Properties
2: Three Different Fourier
Transforms
DFT: X[k] =
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Summary
• MATLAB routines
DSP and Digital Filters (2016-8746)
PN −1
0
kn
x[n]e−j2π N
Fourier Transforms: 2 – 5 / 13
DFT Properties
2: Three Different Fourier
Transforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Summary
• MATLAB routines
DFT: X[k] =
PN −1
0
kn
x[n]e−j2π N
Case 1: x[n] = 0 for n ∈
/ [0, N − 1]
DSP and Digital Filters (2016-8746)
DFT is the same as DTFT at ωk = 2π
N k.
Fourier Transforms: 2 – 5 / 13
DFT Properties
2: Three Different Fourier
Transforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Summary
• MATLAB routines
DFT: X[k] =
jω
DTFT: X(e
PN −1
0
)=
kn
x[n]e−j2π N
P∞
−jωn
x[n]e
−∞
Case 1: x[n] = 0 for n ∈
/ [0, N − 1]
DSP and Digital Filters (2016-8746)
DFT is the same as DTFT at ωk = 2π
N k.
Fourier Transforms: 2 – 5 / 13
DFT Properties
2: Three Different Fourier
Transforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Summary
• MATLAB routines
DFT: X[k] =
jω
DTFT: X(e
PN −1
0
)=
kn
x[n]e−j2π N
P∞
−jωn
x[n]e
−∞
Case 1: x[n] = 0 for n ∈
/ [0, N − 1]
DSP and Digital Filters (2016-8746)
DFT is the same as DTFT at ωk = 2π
N k.
−1
The {ωk } are uniformly spaced from ω = 0 to ω = 2π NN
.
Fourier Transforms: 2 – 5 / 13
DFT Properties
2: Three Different Fourier
Transforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Summary
• MATLAB routines
DFT: X[k] =
jω
DTFT: X(e
PN −1
0
)=
kn
x[n]e−j2π N
P∞
−jωn
x[n]e
−∞
Case 1: x[n] = 0 for n ∈
/ [0, N − 1]
DSP and Digital Filters (2016-8746)
DFT is the same as DTFT at ωk = 2π
N k.
−1
The {ωk } are uniformly spaced from ω = 0 to ω = 2π NN
.
DFT is the z -Transform evaluated at N equally spaced points
around the unit circle beginning at z = 1.
Fourier Transforms: 2 – 5 / 13
DFT Properties
2: Three Different Fourier
Transforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Summary
• MATLAB routines
DFT: X[k] =
jω
DTFT: X(e
PN −1
0
)=
kn
x[n]e−j2π N
P∞
−jωn
x[n]e
−∞
Case 1: x[n] = 0 for n ∈
/ [0, N − 1]
DFT is the same as DTFT at ωk = 2π
N k.
−1
The {ωk } are uniformly spaced from ω = 0 to ω = 2π NN
.
DFT is the z -Transform evaluated at N equally spaced points
around the unit circle beginning at z = 1.
Case 2: x[n] is periodic with period N
DFT equals the normalized DTFT
DSP and Digital Filters (2016-8746)
Fourier Transforms: 2 – 5 / 13
DFT Properties
2: Three Different Fourier
Transforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Summary
• MATLAB routines
DFT: X[k] =
jω
DTFT: X(e
PN −1
0
)=
kn
x[n]e−j2π N
P∞
−jωn
x[n]e
−∞
Case 1: x[n] = 0 for n ∈
/ [0, N − 1]
DFT is the same as DTFT at ωk = 2π
N k.
−1
The {ωk } are uniformly spaced from ω = 0 to ω = 2π NN
.
DFT is the z -Transform evaluated at N equally spaced points
around the unit circle beginning at z = 1.
Case 2: x[n] is periodic with period N
DFT equals the normalized DTFT
X[k] = limK→∞
DSP and Digital Filters (2016-8746)
N
2K+1
× XK (ejωk )
Fourier Transforms: 2 – 5 / 13
DFT Properties
2: Three Different Fourier
Transforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Summary
• MATLAB routines
DFT: X[k] =
jω
DTFT: X(e
PN −1
0
)=
kn
x[n]e−j2π N
P∞
−jωn
x[n]e
−∞
Case 1: x[n] = 0 for n ∈
/ [0, N − 1]
DFT is the same as DTFT at ωk = 2π
N k.
−1
The {ωk } are uniformly spaced from ω = 0 to ω = 2π NN
.
DFT is the z -Transform evaluated at N equally spaced points
around the unit circle beginning at z = 1.
Case 2: x[n] is periodic with period N
DFT equals the normalized DTFT
N
X[k] = limK→∞ 2K+1
× XK (ejωk )
PK
jω
−jωn
where XK (e ) =
−K x[n]e
DSP and Digital Filters (2016-8746)
Fourier Transforms: 2 – 5 / 13
Symmetries
2: Three Different Fourier
Transforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Summary
• MATLAB routines
If x[n] has a special property then X(ejω )and X[k] will have
corresponding properties as shown in the table (and vice versa):
DSP and Digital Filters (2016-8746)
One domain
Other domain
Discrete
Symmetric
Antisymmetric
Real
Imaginary
Real + Symmetric
Real + Antisymmetric
Periodic
Symmetric
Antisymmetric
Conjugate Symmetric
Conjugate Antisymmetric
Real + Symmetric
Imaginary + Antisymmetric
Fourier Transforms: 2 – 6 / 13
Symmetries
2: Three Different Fourier
Transforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Summary
• MATLAB routines
If x[n] has a special property then X(ejω )and X[k] will have
corresponding properties as shown in the table (and vice versa):
One domain
Other domain
Discrete
Symmetric
Antisymmetric
Real
Imaginary
Real + Symmetric
Real + Antisymmetric
Periodic
Symmetric
Antisymmetric
Conjugate Symmetric
Conjugate Antisymmetric
Real + Symmetric
Imaginary + Antisymmetric
Symmetric: x[n] = x[−n]
DSP and Digital Filters (2016-8746)
Fourier Transforms: 2 – 6 / 13
Symmetries
2: Three Different Fourier
Transforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Summary
• MATLAB routines
If x[n] has a special property then X(ejω )and X[k] will have
corresponding properties as shown in the table (and vice versa):
One domain
Other domain
Discrete
Symmetric
Antisymmetric
Real
Imaginary
Real + Symmetric
Real + Antisymmetric
Periodic
Symmetric
Antisymmetric
Conjugate Symmetric
Conjugate Antisymmetric
Real + Symmetric
Imaginary + Antisymmetric
Symmetric: x[n] = x[−n]
X(ejω ) = X(e−jω )
DSP and Digital Filters (2016-8746)
Fourier Transforms: 2 – 6 / 13
Symmetries
2: Three Different Fourier
Transforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Summary
• MATLAB routines
If x[n] has a special property then X(ejω )and X[k] will have
corresponding properties as shown in the table (and vice versa):
One domain
Other domain
Discrete
Symmetric
Antisymmetric
Real
Imaginary
Real + Symmetric
Real + Antisymmetric
Periodic
Symmetric
Antisymmetric
Conjugate Symmetric
Conjugate Antisymmetric
Real + Symmetric
Imaginary + Antisymmetric
Symmetric: x[n] = x[−n]
X(ejω ) = X(e−jω )
X[k] = X[(−k)mod N ] = X[N − k] for k > 0
DSP and Digital Filters (2016-8746)
Fourier Transforms: 2 – 6 / 13
Symmetries
2: Three Different Fourier
Transforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Summary
• MATLAB routines
If x[n] has a special property then X(ejω )and X[k] will have
corresponding properties as shown in the table (and vice versa):
One domain
Other domain
Discrete
Symmetric
Antisymmetric
Real
Imaginary
Real + Symmetric
Real + Antisymmetric
Periodic
Symmetric
Antisymmetric
Conjugate Symmetric
Conjugate Antisymmetric
Real + Symmetric
Imaginary + Antisymmetric
Symmetric: x[n] = x[−n]
X(ejω ) = X(e−jω )
X[k] = X[(−k)mod N ] = X[N − k] for k > 0
Conjugate Symmetric: x[n] = x∗ [−n]
DSP and Digital Filters (2016-8746)
Fourier Transforms: 2 – 6 / 13
Symmetries
2: Three Different Fourier
Transforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Summary
• MATLAB routines
If x[n] has a special property then X(ejω )and X[k] will have
corresponding properties as shown in the table (and vice versa):
One domain
Other domain
Discrete
Symmetric
Antisymmetric
Real
Imaginary
Real + Symmetric
Real + Antisymmetric
Periodic
Symmetric
Antisymmetric
Conjugate Symmetric
Conjugate Antisymmetric
Real + Symmetric
Imaginary + Antisymmetric
Symmetric: x[n] = x[−n]
X(ejω ) = X(e−jω )
X[k] = X[(−k)mod N ] = X[N − k] for k > 0
Conjugate Symmetric: x[n] = x∗ [−n]
Conjugate Antisymmetric: x[n] = −x∗ [−n]
DSP and Digital Filters (2016-8746)
Fourier Transforms: 2 – 6 / 13
Parseval’s Theorem
2: Three Different Fourier
Transforms
Fourier transforms preserve “energy”
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Summary
• MATLAB routines
DSP and Digital Filters (2016-8746)
Fourier Transforms: 2 – 7 / 13
Parseval’s Theorem
2: Three Different Fourier
Transforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Summary
• MATLAB routines
Fourier transforms preserve “energy”
CTFT
DSP and Digital Filters (2016-8746)
R
2
|x(t)| dt =
1
2π
R
|X(jΩ)|2 dΩ
Fourier Transforms: 2 – 7 / 13
Parseval’s Theorem
2: Three Different Fourier
Transforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Summary
• MATLAB routines
Fourier transforms preserve “energy”
CTFT
DTFT
DSP and Digital Filters (2016-8746)
2
R
1
|x(t)| dt = 2π |X(jΩ)|2 dΩ
Rπ P∞
2
1
jω 2
−∞ |x[n]| = 2π −π X(e ) dω
R
Fourier Transforms: 2 – 7 / 13
Parseval’s Theorem
2: Three Different Fourier
Transforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Summary
• MATLAB routines
Fourier transforms preserve “energy”
CTFT
DTFT
DFT
DSP and Digital Filters (2016-8746)
2
R
1
|x(t)| dt = 2π |X(jΩ)|2 dΩ
Rπ P∞
2
1
jω 2
−∞ |x[n]| = 2π −π X(e ) dω
PN −1
PN −1
2
2
1
|x[n]|
=
|X[k]|
0
0
N
R
Fourier Transforms: 2 – 7 / 13
Parseval’s Theorem
2: Three Different Fourier
Transforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Summary
• MATLAB routines
Fourier transforms preserve “energy”
CTFT
DTFT
DFT
2
R
1
|x(t)| dt = 2π |X(jΩ)|2 dΩ
Rπ P∞
2
1
jω 2
−∞ |x[n]| = 2π −π X(e ) dω
PN −1
PN −1
2
2
1
|x[n]|
=
|X[k]|
0
0
N
R
More generally, they actually preserve complex inner products:
DSP and Digital Filters (2016-8746)
PN −1
0
x[n]y [n] =
∗
1
N
PN −1
0
X[k]Y ∗ [k]
Fourier Transforms: 2 – 7 / 13
Parseval’s Theorem
2: Three Different Fourier
Transforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Summary
• MATLAB routines
Fourier transforms preserve “energy”
CTFT
DTFT
DFT
2
R
1
|x(t)| dt = 2π |X(jΩ)|2 dΩ
Rπ P∞
2
1
jω 2
−∞ |x[n]| = 2π −π X(e ) dω
PN −1
PN −1
2
2
1
|x[n]|
=
|X[k]|
0
0
N
R
More generally, they actually preserve complex inner products:
PN −1
0
x[n]y [n] =
∗
1
N
PN −1
0
X[k]Y ∗ [k]
Unitary matrix viewpoint for DFT:
If we regard x and X as vectors, then X = Fx where F is
kn
a symmetric matrix defined by fk+1,n+1 = e−j2π N .
DSP and Digital Filters (2016-8746)
Fourier Transforms: 2 – 7 / 13
Parseval’s Theorem
2: Three Different Fourier
Transforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Summary
• MATLAB routines
Fourier transforms preserve “energy”
CTFT
DTFT
DFT
2
R
1
|x(t)| dt = 2π |X(jΩ)|2 dΩ
Rπ P∞
2
1
jω 2
−∞ |x[n]| = 2π −π X(e ) dω
PN −1
PN −1
2
2
1
|x[n]|
=
|X[k]|
0
0
N
R
More generally, they actually preserve complex inner products:
PN −1
0
x[n]y [n] =
∗
1
N
PN −1
0
X[k]Y ∗ [k]
Unitary matrix viewpoint for DFT:
If we regard x and X as vectors, then X = Fx where F is
kn
a symmetric matrix defined by fk+1,n+1 = e−j2π N .
1 H
The inverse DFT matrix is F−1 = N
F
equivalently, G = √1 F is a unitary matrix with GH G = I.
N
DSP and Digital Filters (2016-8746)
Fourier Transforms: 2 – 7 / 13
Convolution
2: Three Different Fourier
Transforms
DTFT: Convolution → Product
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Summary
• MATLAB routines
DSP and Digital Filters (2016-8746)
Fourier Transforms: 2 – 8 / 13
Convolution
2: Three Different Fourier
Transforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Summary
• MATLAB routines
DTFT: Convolution → Product
x[n] = g[n] ∗ h[n]=
DSP and Digital Filters (2016-8746)
P∞
k=−∞ g[k]h[n
− k]
Fourier Transforms: 2 – 8 / 13
Convolution
2: Three Different Fourier
Transforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Summary
• MATLAB routines
DTFT: Convolution → Product
P
x[n] = g[n] ∗ h[n]= ∞
k=−∞ g[k]h[n − k]
⇒ X(ejω ) = G(ejω )H(ejω )
DSP and Digital Filters (2016-8746)
Fourier Transforms: 2 – 8 / 13
Convolution
2: Three Different Fourier
Transforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Summary
• MATLAB routines
DTFT: Convolution → Product
P
x[n] = g[n] ∗ h[n]= ∞
k=−∞ g[k]h[n − k]
⇒ X(ejω ) = G(ejω )H(ejω )
DFT: Circular convolution→ Product
x[n] = g[n] ⊛N h[n]=
DSP and Digital Filters (2016-8746)
PN −1
k=0
g[k]h[(n − k)modN ]
Fourier Transforms: 2 – 8 / 13
Convolution
2: Three Different Fourier
Transforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Summary
• MATLAB routines
DTFT: Convolution → Product
P
x[n] = g[n] ∗ h[n]= ∞
k=−∞ g[k]h[n − k]
⇒ X(ejω ) = G(ejω )H(ejω )
DFT: Circular convolution→ Product
PN −1
x[n] = g[n] ⊛N h[n]=
⇒ X[k] = G[k]H[k]
DSP and Digital Filters (2016-8746)
k=0
g[k]h[(n − k)modN ]
Fourier Transforms: 2 – 8 / 13
Convolution
2: Three Different Fourier
Transforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Summary
• MATLAB routines
DTFT: Convolution → Product
P
x[n] = g[n] ∗ h[n]= ∞
k=−∞ g[k]h[n − k]
⇒ X(ejω ) = G(ejω )H(ejω )
DFT: Circular convolution→ Product
PN −1
x[n] = g[n] ⊛N h[n]=
⇒ X[k] = G[k]H[k]
g[n] :
DSP and Digital Filters (2016-8746)
k=0
g[k]h[(n − k)modN ]
h[n] :
Fourier Transforms: 2 – 8 / 13
Convolution
2: Three Different Fourier
Transforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Summary
• MATLAB routines
DTFT: Convolution → Product
P
x[n] = g[n] ∗ h[n]= ∞
k=−∞ g[k]h[n − k]
⇒ X(ejω ) = G(ejω )H(ejω )
DFT: Circular convolution→ Product
PN −1
x[n] = g[n] ⊛N h[n]=
⇒ X[k] = G[k]H[k]
g[n] :
DSP and Digital Filters (2016-8746)
h[n] :
k=0
g[k]h[(n − k)modN ]
g[n] ∗ h[n] :
Fourier Transforms: 2 – 8 / 13
Convolution
2: Three Different Fourier
Transforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Summary
• MATLAB routines
DTFT: Convolution → Product
P
x[n] = g[n] ∗ h[n]= ∞
k=−∞ g[k]h[n − k]
⇒ X(ejω ) = G(ejω )H(ejω )
DFT: Circular convolution→ Product
PN −1
x[n] = g[n] ⊛N h[n]=
⇒ X[k] = G[k]H[k]
g[n] :
DSP and Digital Filters (2016-8746)
h[n] :
k=0
g[k]h[(n − k)modN ]
g[n] ∗ h[n] :
g[n] ⊛3 h[n]
Fourier Transforms: 2 – 8 / 13
Convolution
2: Three Different Fourier
Transforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Summary
• MATLAB routines
DTFT: Convolution → Product
P
x[n] = g[n] ∗ h[n]= ∞
k=−∞ g[k]h[n − k]
⇒ X(ejω ) = G(ejω )H(ejω )
DFT: Circular convolution→ Product
PN −1
x[n] = g[n] ⊛N h[n]=
⇒ X[k] = G[k]H[k]
k=0
g[k]h[(n − k)modN ]
DTFT: Product→ Circular Convolution ÷2π
y[n] = g[n]h[n]
g[n] :
DSP and Digital Filters (2016-8746)
h[n] :
g[n] ∗ h[n] :
g[n] ⊛3 h[n]
Fourier Transforms: 2 – 8 / 13
Convolution
2: Three Different Fourier
Transforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Summary
• MATLAB routines
DTFT: Convolution → Product
P
x[n] = g[n] ∗ h[n]= ∞
k=−∞ g[k]h[n − k]
⇒ X(ejω ) = G(ejω )H(ejω )
DFT: Circular convolution→ Product
PN −1
x[n] = g[n] ⊛N h[n]=
⇒ X[k] = G[k]H[k]
k=0
g[k]h[(n − k)modN ]
DTFT: Product→ Circular Convolution ÷2π
1
2π
y[n] = g[n]h[n]
1
⇒ Y (ejω ) = 2π
G(ejω ) ⊛π H(ejω ) =
Rπ
G(ejθ )H(ej(ω−θ) )dθ
−π
g[n] :
DSP and Digital Filters (2016-8746)
h[n] :
g[n] ∗ h[n] :
g[n] ⊛3 h[n]
Fourier Transforms: 2 – 8 / 13
Convolution
2: Three Different Fourier
Transforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Summary
• MATLAB routines
DTFT: Convolution → Product
P
x[n] = g[n] ∗ h[n]= ∞
k=−∞ g[k]h[n − k]
⇒ X(ejω ) = G(ejω )H(ejω )
DFT: Circular convolution→ Product
PN −1
x[n] = g[n] ⊛N h[n]=
⇒ X[k] = G[k]H[k]
k=0
g[k]h[(n − k)modN ]
DTFT: Product→ Circular Convolution ÷2π
1
2π
y[n] = g[n]h[n]
1
⇒ Y (ejω ) = 2π
G(ejω ) ⊛π H(ejω ) =
Rπ
G(ejθ )H(ej(ω−θ) )dθ
−π
DFT: Product→ Circular Convolution ÷N
y[n] = g[n]h[n]
g[n] :
DSP and Digital Filters (2016-8746)
h[n] :
g[n] ∗ h[n] :
g[n] ⊛3 h[n]
Fourier Transforms: 2 – 8 / 13
Convolution
2: Three Different Fourier
Transforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Summary
• MATLAB routines
DTFT: Convolution → Product
P
x[n] = g[n] ∗ h[n]= ∞
k=−∞ g[k]h[n − k]
⇒ X(ejω ) = G(ejω )H(ejω )
DFT: Circular convolution→ Product
PN −1
x[n] = g[n] ⊛N h[n]=
⇒ X[k] = G[k]H[k]
k=0
g[k]h[(n − k)modN ]
DTFT: Product→ Circular Convolution ÷2π
1
2π
y[n] = g[n]h[n]
1
⇒ Y (ejω ) = 2π
G(ejω ) ⊛π H(ejω ) =
Rπ
G(ejθ )H(ej(ω−θ) )dθ
−π
DFT: Product→ Circular Convolution ÷N
y[n] = g[n]h[n]
⇒ Y [k] = N1 G[k] ⊛N H[k]
g[n] :
DSP and Digital Filters (2016-8746)
h[n] :
g[n] ∗ h[n] :
g[n] ⊛3 h[n]
Fourier Transforms: 2 – 8 / 13
Sampling Process
Time
Analog
DSP and Digital Filters (2016-8746)
Time
Frequency
CTFT
−→
Fourier Transforms: 2 – 9 / 13
Sampling Process
Time
Time
CTFT
−→
Analog
Low Pass
Filter
Frequency
*
DSP and Digital Filters (2016-8746)
→
CTFT
−→
Fourier Transforms: 2 – 9 / 13
Sampling Process
Time
Time
Frequency
CTFT
−→
Analog
Low Pass
Filter
*
→
−→
Sample
×
→
−→
DSP and Digital Filters (2016-8746)
CTFT
DTFT
Fourier Transforms: 2 – 9 / 13
Sampling Process
Time
Time
Frequency
CTFT
−→
Analog
Low Pass
Filter
*
→
−→
Sample
×
→
−→
Window
×
→
−→
DSP and Digital Filters (2016-8746)
CTFT
DTFT
DTFT
Fourier Transforms: 2 – 9 / 13
Sampling Process
Time
Time
Frequency
CTFT
−→
Analog
Low Pass
Filter
*
→
−→
Sample
×
→
−→
Window
×
→
−→
DFT
DSP and Digital Filters (2016-8746)
CTFT
DTFT
DTFT
DFT
−→
Fourier Transforms: 2 – 9 / 13
Zero-Padding
2: Three Different Fourier
Transforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Summary
• MATLAB routines
Zero padding means added extra zeros onto the end of x[n] before
performing the DFT.
DSP and Digital Filters (2016-8746)
Time x[n]
Windowed
Signal
Fourier Transforms: 2 – 10 / 13
Zero-Padding
2: Three Different Fourier
Transforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Summary
• MATLAB routines
Zero padding means added extra zeros onto the end of x[n] before
performing the DFT.
Time x[n]
Windowed
Signal
With zeropadding
DSP and Digital Filters (2016-8746)
Fourier Transforms: 2 – 10 / 13
Zero-Padding
2: Three Different Fourier
Transforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Summary
• MATLAB routines
Zero padding means added extra zeros onto the end of x[n] before
performing the DFT.
Time x[n]
Frequency |X[k]|
Windowed
Signal
With zeropadding
DSP and Digital Filters (2016-8746)
Fourier Transforms: 2 – 10 / 13
Zero-Padding
2: Three Different Fourier
Transforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Summary
• MATLAB routines
Zero padding means added extra zeros onto the end of x[n] before
performing the DFT.
Time x[n]
Frequency |X[k]|
Windowed
Signal
With zeropadding
DSP and Digital Filters (2016-8746)
Fourier Transforms: 2 – 10 / 13
Zero-Padding
2: Three Different Fourier
Transforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Summary
• MATLAB routines
Zero padding means added extra zeros onto the end of x[n] before
performing the DFT.
Time x[n]
Frequency |X[k]|
Windowed
Signal
With zeropadding
• Zero-padding causes the DFT to evaluate the DTFT at more values
of ωk . Denser frequency samples.
DSP and Digital Filters (2016-8746)
Fourier Transforms: 2 – 10 / 13
Zero-Padding
2: Three Different Fourier
Transforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Summary
• MATLAB routines
Zero padding means added extra zeros onto the end of x[n] before
performing the DFT.
Time x[n]
Frequency |X[k]|
Windowed
Signal
With zeropadding
• Zero-padding causes the DFT to evaluate the DTFT at more values
of ωk . Denser frequency samples.
• Width of the peaks remains constant: determined by the length and
shape of the window.
DSP and Digital Filters (2016-8746)
Fourier Transforms: 2 – 10 / 13
Zero-Padding
2: Three Different Fourier
Transforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Summary
• MATLAB routines
Zero padding means added extra zeros onto the end of x[n] before
performing the DFT.
Time x[n]
Frequency |X[k]|
Windowed
Signal
With zeropadding
• Zero-padding causes the DFT to evaluate the DTFT at more values
of ωk . Denser frequency samples.
• Width of the peaks remains constant: determined by the length and
•
shape of the window.
Smoother graph but increased frequency resolution is an illusion.
DSP and Digital Filters (2016-8746)
Fourier Transforms: 2 – 10 / 13
Phase Unwrapping
2: Three Different Fourier
Transforms
Phase of a DTFT is only defined to within an integer multiple of 2π .
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Summary
• MATLAB routines
DSP and Digital Filters (2016-8746)
x[n]
|X[k]|
Fourier Transforms: 2 – 11 / 13
Phase Unwrapping
2: Three Different Fourier
Transforms
Phase of a DTFT is only defined to within an integer multiple of 2π .
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Summary
• MATLAB routines
x[n]
|X[k]|
∠X[k]
DSP and Digital Filters (2016-8746)
Fourier Transforms: 2 – 11 / 13
Phase Unwrapping
2: Three Different Fourier
Transforms
Phase of a DTFT is only defined to within an integer multiple of 2π .
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Summary
• MATLAB routines
x[n]
|X[k]|
∠X[k]
Phase unwrapping adds multiples of 2π onto each ∠X[k] to make the
phase as continuous as possible.
DSP and Digital Filters (2016-8746)
Fourier Transforms: 2 – 11 / 13
Phase Unwrapping
2: Three Different Fourier
Transforms
Phase of a DTFT is only defined to within an integer multiple of 2π .
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Summary
• MATLAB routines
x[n]
|X[k]|
∠X[k]
∠X[k] unwrapped
Phase unwrapping adds multiples of 2π onto each ∠X[k] to make the
phase as continuous as possible.
DSP and Digital Filters (2016-8746)
Fourier Transforms: 2 – 11 / 13
Summary
2: Three Different Fourier
Transforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Summary
• MATLAB routines
• Three types: CTFT, DTFT, DFT
DSP and Digital Filters (2016-8746)
Fourier Transforms: 2 – 12 / 13
Summary
2: Three Different Fourier
Transforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Summary
• MATLAB routines
• Three types: CTFT, DTFT, DFT
DSP and Digital Filters (2016-8746)
◦ DTFT = CTFT of continuous signal × impulse train
Fourier Transforms: 2 – 12 / 13
Summary
2: Three Different Fourier
Transforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Summary
• MATLAB routines
• Three types: CTFT, DTFT, DFT
DSP and Digital Filters (2016-8746)
◦ DTFT = CTFT of continuous signal × impulse train
◦ DFT = DTFT of periodic or finite support signal
Fourier Transforms: 2 – 12 / 13
Summary
2: Three Different Fourier
Transforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Summary
• MATLAB routines
• Three types: CTFT, DTFT, DFT
DSP and Digital Filters (2016-8746)
◦ DTFT = CTFT of continuous signal × impulse train
◦ DFT = DTFT of periodic or finite support signal
•
DFT is a scaled unitary transform
Fourier Transforms: 2 – 12 / 13
Summary
2: Three Different Fourier
Transforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Summary
• MATLAB routines
• Three types: CTFT, DTFT, DFT
◦ DTFT = CTFT of continuous signal × impulse train
◦ DFT = DTFT of periodic or finite support signal
•
DFT is a scaled unitary transform
• DTFT: Convolution → Product; Product → Circular Convolution
DSP and Digital Filters (2016-8746)
Fourier Transforms: 2 – 12 / 13
Summary
2: Three Different Fourier
Transforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Summary
• MATLAB routines
• Three types: CTFT, DTFT, DFT
◦ DTFT = CTFT of continuous signal × impulse train
◦ DFT = DTFT of periodic or finite support signal
•
DFT is a scaled unitary transform
• DTFT: Convolution → Product; Product → Circular Convolution
• DFT: Product ↔ Circular Convolution
DSP and Digital Filters (2016-8746)
Fourier Transforms: 2 – 12 / 13
Summary
2: Three Different Fourier
Transforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Summary
• MATLAB routines
• Three types: CTFT, DTFT, DFT
◦ DTFT = CTFT of continuous signal × impulse train
◦ DFT = DTFT of periodic or finite support signal
•
DFT is a scaled unitary transform
• DTFT: Convolution → Product; Product → Circular Convolution
• DFT: Product ↔ Circular Convolution
• DFT: Zero Padding → Denser freq sampling but same resolution
DSP and Digital Filters (2016-8746)
Fourier Transforms: 2 – 12 / 13
Summary
2: Three Different Fourier
Transforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Summary
• MATLAB routines
• Three types: CTFT, DTFT, DFT
◦ DTFT = CTFT of continuous signal × impulse train
◦ DFT = DTFT of periodic or finite support signal
•
DFT is a scaled unitary transform
• DTFT: Convolution → Product; Product → Circular Convolution
• DFT: Product ↔ Circular Convolution
• DFT: Zero Padding → Denser freq sampling but same resolution
• Phase is only defined to within a multiple of 2π .
DSP and Digital Filters (2016-8746)
Fourier Transforms: 2 – 12 / 13
Summary
2: Three Different Fourier
Transforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Summary
• MATLAB routines
• Three types: CTFT, DTFT, DFT
◦ DTFT = CTFT of continuous signal × impulse train
◦ DFT = DTFT of periodic or finite support signal
•
DFT is a scaled unitary transform
• DTFT: Convolution → Product; Product → Circular Convolution
• DFT: Product ↔ Circular Convolution
• DFT: Zero Padding → Denser freq sampling but same resolution
• Phase is only defined to within a multiple of 2π .
• Whenever you integrate over frequency you need a scale factor
1
1
for CTFT and DTFT or N
for DFT
◦ 2π
◦ e.g. Inverse transform, Parseval, frequency domain convolution
DSP and Digital Filters (2016-8746)
Fourier Transforms: 2 – 12 / 13
Summary
2: Three Different Fourier
Transforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Summary
• MATLAB routines
• Three types: CTFT, DTFT, DFT
◦ DTFT = CTFT of continuous signal × impulse train
◦ DFT = DTFT of periodic or finite support signal
•
DFT is a scaled unitary transform
• DTFT: Convolution → Product; Product → Circular Convolution
• DFT: Product ↔ Circular Convolution
• DFT: Zero Padding → Denser freq sampling but same resolution
• Phase is only defined to within a multiple of 2π .
• Whenever you integrate over frequency you need a scale factor
1
1
for CTFT and DTFT or N
for DFT
◦ 2π
◦ e.g. Inverse transform, Parseval, frequency domain convolution
For further details see Mitra: 3 & 5.
DSP and Digital Filters (2016-8746)
Fourier Transforms: 2 – 12 / 13
MATLAB routines
2: Three Different Fourier
Transforms
• Fourier Transforms
• Convergence of DTFT
• DTFT Properties
• DFT Properties
• Symmetries
• Parseval’s Theorem
• Convolution
• Sampling Process
• Zero-Padding
• Phase Unwrapping
• Summary
• MATLAB routines
fft, ifft
fftshift
conv
x[n]⊛y[n]
unwrap
DSP and Digital Filters (2016-8746)
DFT with optional zero-padding
swap the two halves of a vector
convolution or polynomial multiplication (not circular)
real(ifft(fft(x).*fft(y)))
remove 2π jumps from phase spectrum
Fourier Transforms: 2 – 13 / 13