lec 8: february 7th, 2017 sampling and reconstructionese531/spring2017/handouts/lec8.pdf · lec 8:...
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ESE 531: Digital Signal Processing
Lec 8: February 7th, 2017 Sampling and Reconstruction
Penn ESE 531 Spring 2017 - Khanna
Lecture Outline
! Review " Ideal sampling " Frequency response of sampled signal " Reconstruction " Anti-aliasing filtering
! DT processing of CT signals ! CT processing of DT signals (why??)
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Last Time…
Sampling, Frequency Response of Sampled Signal, Reconstruction, Anti-aliasing filtering
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DSP System
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Ideal Sampling Model
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Frequency Domain Analysis
! How is x[n] related to xs(t) in frequency domain?
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Frequency Domain Analysis
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Xs (ejΩ)
Ωs
2⋅T = π
ω =ΩT
Frequency Domain Analysis
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Xs (ejΩ)
Ωs
2⋅T = π
ω =ΩT
Reconstruction of Bandlimited Signals
! Nyquist Sampling Theorem: Suppose xc(t) is bandlimited. I.e.
! If Ωs≥2ΩN, then xc(t) can be uniquely determined from its samples x[n]=xc(nT)
! Bandlimitedness is the key to uniqueness
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Mulitiple signals go through the samples, but only one is
bandlimited within our sampling band
Reconstruction in Frequency Domain
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Reconstruction in Time Domain
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*
= The sum of “sincs”
gives xr(t) # unique signal that is
bandlimited by sampling bandwidth
Aliasing
! If ΩN>Ωs/2, xr(t) an aliased version of xc(t)
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Anti-Aliasing Filter
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Anti-Aliasing Filter
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MRI aliasing example
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MRI anti-aliasing example
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MRI anti-aliasing example
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Reconstruction in Frequency Domain
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Different T?
Discrete-Time Processing of Continuous Time
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x[n] y[n]
X (e jω ) = 1T
Xc jωT−2πkT
⎛
⎝⎜
⎞
⎠⎟
⎡
⎣⎢
⎤
⎦⎥
k=−∞
∞
∑
T T
Discrete-Time Processing of Continuous Time
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x[n] y[n]
X (e jω ) = 1T
Xc jωT−2πkT
⎛
⎝⎜
⎞
⎠⎟
⎡
⎣⎢
⎤
⎦⎥
k=−∞
∞
∑
T T
Xc ( j(Ω− kΩs ))
Frequency Domain Analysis
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Xs (ejΩ)
Ωs
2⋅T = π
ω =ΩT
Discrete-Time Processing of Continuous Time
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x[n] y[n]
X (e jω ) = 1T
Xc jωT−2πkT
⎛
⎝⎜
⎞
⎠⎟
⎡
⎣⎢
⎤
⎦⎥
k=−∞
∞
∑
T T
yr (t) = y[n] sin[π (t − nT ) /T ]π (t − nT ) /Tn=−∞
∞
∑
Reconstruction in Frequency Domain
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Reconstruction in Time Domain
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*
= The sum of “sincs”
gives xr(t) # unique signal that is
bandlimited by sampling bandwidth
Discrete-Time Processing of Continuous Time
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x[n] y[n]
X (e jω ) = 1T
Xc jωT−2πkT
⎛
⎝⎜
⎞
⎠⎟
⎡
⎣⎢
⎤
⎦⎥
k=−∞
∞
∑
T T
yr (t) = y[n] sin[π (t − nT ) /T ]π (t − nT ) /Tn=−∞
∞
∑
Sum of scaled shifted sincs
Discrete-Time Processing of Continuous Time
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x[n] y[n]
X (e jω ) = 1T
Xc jωT−2πkT
⎛
⎝⎜
⎞
⎠⎟
⎡
⎣⎢
⎤
⎦⎥
k=−∞
∞
∑
T T
yr (t) = y[n] sin[π (t − nT ) /T ]π (t − nT ) /Tn=−∞
∞
∑
! If h[n] is LTI, H(ejω) exists " Is the whole system from xc(t)#yc(t) LTI?
Discrete-Time Processing of Continuous Time
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x[n] y[n]
X (e jω ) = 1T
Xc jωT−2πkT
⎛
⎝⎜
⎞
⎠⎟
⎡
⎣⎢
⎤
⎦⎥
k=−∞
∞
∑
T T
yr (t) = y[n] sin[π (t − nT ) /T ]π (t − nT ) /Tn=−∞
∞
∑
! If xc(t) is bandlimited by Ωs/T=π/T, then, Yr ( jΩ)
Xc ( jΩ)= Heff ( jΩ) =
H (e jω )ω=ΩT
Ω <Ωs /T
0 else
⎧⎨⎪
⎩⎪
Example
! Consider the following system
! Where
! What is the effective frequency response of the system? What happens to a signal bandlimited by ΩN?
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H (e jω ) =1 ω <ωc
0 ωc < ω ≤ π
⎧
⎨⎪
⎩⎪
Impulse Invariance
! Want to implement continuous-time system in discrete-time
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Impulse Invariance
! With Hc(jΩ) bandlimited, choose
! With the further requirement that T be chosen such that
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H (e jω ) = Hc ( jω /T ), ω < π
Hc ( jΩ) = 0, Ω ≥ π /T
Impulse Invariance
! With Hc(jΩ) bandlimited, choose
! With the further requirement that T be chosen such that
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H (e jω ) = Hc ( jω /T ), ω < π
Hc ( jΩ) = 0, Ω ≥ π /T
h[n]=Thc (nT )
Continuous-Time Processing of Discrete-Time
! Useful to interpret DT systems with no simple interpretation in discrete time
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yc (t) = y[n] sin[π (t − nT ) /T ]π (t − nT ) /Tn=−∞
∞
∑
xc (t) = x[n] sin[π (t − nT ) /T ]π (t − nT ) /Tn=−∞
∞
∑
Continuous-Time Processing of Discrete-Time
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Xc ( jΩ) =TX (e jΩT ) Ω < π /T
0 else
⎧⎨⎪
⎩⎪
Continuous-Time Processing of Discrete-Time
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Xc ( jΩ) =TX (e jΩT ) Ω < π /T
0 else
⎧⎨⎪
⎩⎪
Yc ( jΩ) = Hc ( jΩ)Xc ( jΩ)Also bandlimited
Continuous-Time Processing of Discrete-Time
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Yc ( jΩ) = Hc ( jΩ)Xc ( jΩ)Also bandlimited
Y (e jω ) = 1T
Yc j Ω− kΩs( )⎡⎣
⎤⎦
k=−∞
∞
∑Ω=ω /T
=1TYc ( jΩ)
Ω=ω /T
Continuous-Time Processing of Discrete-Time
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Yc ( jΩ) = Hc ( jΩ)Xc ( jΩ) Y (e jω ) = 1TYc ( jΩ)
Ω=ω /T
Y (e jω ) = 1THc ( jΩ) Ω=ω /T Xc ( jΩ) Ω=ω /T
=1THc ( jΩ) Ω=ω /T (TX (e
jω )) = H (e jω )X (e jω ) ω < π
Example
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Example: Non-integer Delay
! What is the time domain operation when Δ is non-integer? I.e Δ=1/2
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Reminder: Properties of the DTFT
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! Time Reversal:
! Time/Freq Shifting:
x[n]↔ X (e jω )x[−n]↔ X (e− jω )
If x[n] real
x[−n]↔ X *(e− jω )
x[n]↔ X (e jω )
x[n− nd ]↔ e− jωnd X (e jω )
e jω0nx[n]↔ X (e j(ω−ω0 ) )
Example: Non-integer Delay
! What is the time domain operation when Δ is non-integer? I.e Δ=1/2
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Example: Non-integer Delay
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Example: Non-integer Delay
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Example: Non-integer Delay
! The block diagram is for interpretation/analysis only
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yc (t) = xc (t −TΔ)
Example: Non-integer Delay
! The block diagram is for interpretation/analysis only
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yc (t) = xc (t −TΔ) y[n]= yc (nT ) = xc (nt −TΔ)
= x[k]k∑ sinc t − kT −TΔ
T
⎛
⎝⎜
⎞
⎠⎟t=nT
= x[k]k∑ sinc n− k −Δ( )
! My delay system has an impulse response of a sinc with a continuous time delay
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Example: Non-integer Delay
! My delay system has an impulse response of a sinc with a continuous time delay
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Example: Non-integer Delay
! My delay system has an impulse response of a sinc with a continuous time delay
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Example: Non-integer Delay
! My delay system has an impulse response of a sinc with a continuous time delay
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Example: Non-integer Delay
Big Ideas
! Sampling and reconstruction " Rely on bandlimitedness for unique reconstruction
! CT processing of DT " Effectively LTI if no aliasing
! DT processing of CT " Always LTI " Useful for interpretation
! Changing the sampling rates next time " Upsampling, downsampling
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Admin
! HW 2 due Friday
! Ahead of schedule " Watch course calendar online for changes
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