ESE 531: Digital Signal Processing Lec 8: February 12th, 2019 Sampling and Reconstruction
Lecture Outline! Review " Ideal sampling " Frequency response of sampled signal " Reconstruction " Anti-aliasing filtering! DT processing of CT signals " Impulse Invariance! CT processing of DT signals (why??) 2
Last Time Sampling, Frequency Response of Sampled Signal, Reconstruction, Anti-aliasing filtering 3
DSP System 4
Ideal Sampling Model 5
Frequency Domain Analysis! How is x[n] related to x s (t) in frequency domain? 6
Frequency Domain Analysis X (e jω ) ω = ΩT ω Ω s 2 T = π 7
Frequency Domain Analysis X (e jω ) ω = ΩT ω Ω s 2 T = π 8
Reconstruction of Bandlimited Signals! Nyquist Sampling Theorem: Suppose x c (t) is bandlimited. I.e.! If Ω s 2Ω N, then x c (t) can be uniquely determined from its samples x[n]=x c (nt)! Bandlimitedness is the key to uniqueness Mulitiple signals go through the samples, but only one is bandlimited within our sampling band 9
Reconstruction in Frequency Domain 10
Reconstruction in Time Domain * = The sum of sincs gives x r (t) # unique signal that is bandlimited by sampling bandwidth 11
Aliasing! If Ω N >Ω s /2, x r (t) an aliased version of x c (t) 12
Anti-Aliasing Filter X C ( jω) 1 1/T X S ( jω) Ω S /2 -Ω N Ω N Ω N X C ( jω)x LP ( jω) 1 1/T X S ( jω) Ω S /2 -Ω N Ω N Ω N Ω S /2 13
Reconstruction in Frequency Domain Different T? 14
Signal Processing! Use theory of sampling (C/D) and reconstruction (D/C) to implement signal processing! Two cases: " Discrete-time processing of continuous-time signals x[n] y[n] 15
Signal Processing! Use theory of sampling (C/D) and reconstruction (D/C) to implement signal processing! Two cases: " Discrete-time processing of continuous-time signals x[n] y[n] " Continuous-time processing of discrete-time signals 16
Discrete-Time Processing of Continuous Time x[n] y[n] T T 17
Discrete-Time Processing of Continuous Time x[n] y[n] T T X (e jω ) = 1 T k= X c ω j T 2πk T X (e jω ) ω Ω s 2 T = π 18
Discrete-Time Processing of Continuous Time x[n] y[n] T T X (e jω ) = 1 T k= X c ω j T 2πk T Sum of scaled shifted sincs n= y r (t) = y[n] sin[π (t nt ) / T ] π (t nt ) / T 19
Discrete-Time Processing of Continuous Time x[n] y[n] T T X (e jω ) = 1 T k= X c ω j T 2πk T sin[π (t nt ) / T ] y r (t) = y[n] π (t nt ) / T n=! If h[n]/h(e jω ) is LTI " Is the whole system from x c (t)#y c (t) LTI? 20
Discrete-Time Processing of Continuous Time x[n] y[n] T T X (e jω ) = 1 T k= X c ω j T 2πk T sin[π (t nt ) / T ] y r (t) = y[n] π (t nt ) / T n=! If x c (t) is bandlimited by Ω s /T=π/T, then, Y r ( jω) X c ( jω) = H ( jω) = H(e jω ) Ω < Ω s / T ω=ωt eff 0 else 21
Example 1! Consider the following system T T! Where! What is the effective frequency response of the system? What happens to a signal bandlimited by Ω N? H(e jω ) = 1 ω < ω c 0 ω c < ω π 22
Example 2! DT implementation of an ideal CT bandlimited differentiator T T! The ideal CT differentiator is defined by y C (t) = d dt [x (t)] C! With corresponding H C ( jω) = jω 23
Impulse Invariance! Want to implement continuous-time system 24
Impulse Invariance! Want to implement continuous-time system! in discrete-time 25
Impulse Invariance! With H c (jω) bandlimited, choose H(e jω ) = H c ( jω) ω, ω < π Ω= T! With the further requirement that T be chosen such that H c ( jω) = 0, Ω π / T 26
Impulse Invariance! With H c (jω) bandlimited, choose H(e jω ) = H c ( jω) ω, ω < π Ω= T! With the further requirement that T be chosen such that H c ( jω) = 0, Ω π / T h[n] = Th c (nt ) 27
Impulse Invariance! Let, h[n] = h c (nt ) 28
Impulse Invariance! Let, h[n] = h c (nt )! If sampling at Nyquist Rate then H(e jω ) = 1 T k= H c ω j T 2πk T 29
Impulse Invariance! Let, h[n] = h c (nt )! If sampling at Nyquist Rate then H(e jω ) = 1 T k= H c H c ( jω) = 0, ω j T 2πk T Ω π / T H(e jω ) = 1 T H c j ω T, ω <π 30
Impulse Invariance! Let, h[n] = T h c (nt )! If sampling at Nyquist Rate then H(e jω T ) = 1 T k= H c H c ( jω) = 0, ω j T 2πk T Ω π / T H(e jω T ) = 1 T H c j ω T, ω <π 31
Impulse Invariance! Want to implement continuous-time system in discrete-time h[n] = Th c (nt ) 32
Example 3: DT Lowpass Filter! We wish to implement a lowpass filter with cutoff frequency Ω c on continuous time signal in discrete time with the following system 33
Continuous-Time Processing of Discrete-Time! Useful to interpret DT systems with no simple interpretation in discrete time Is the effective H(e jω ) LTI? 34
Continuous-Time Processing of Discrete-Time X c ( jω) = TX (e jωt ) Ω < π / T 0 else 35
Continuous-Time Processing of Discrete-Time X c ( jω) = TX (e jωt ) Ω < π / T 0 else Also bandlimited Y c ( jω) = H c ( jω)x c ( jω) 36
Continuous-Time Processing of Discrete-Time Y c ( jω) = H c ( jω)x c ( jω) Y (e jω ) = 1 T Y c k= ( ) j Ω kω s Ω=ω/T = 1 T Y c ( jω) Ω=ω/T 37
Continuous-Time Processing of Discrete-Time Y c ( jω) = H c ( jω)x c ( jω) Y (e jω ) = 1 T Y c k= ( ) j Ω kω s Ω=ω/T = 1 T Y c ( jω) Ω=ω/T 38
Continuous-Time Processing of Discrete-Time Y c ( jω) = H c ( jω)x c ( jω) Y (e jω ) = 1 T Y c ( jω) Ω=ω/T 39
Continuous-Time Processing of Discrete-Time Y c ( jω) = H c ( jω)x c ( jω) Y (e jω ) = 1 T Y c ( jω) Ω=ω/T Y (e jω ) = 1 T H c ( jω) Ω=ω/T X c ( jω) Ω=ω/T = 1 T H c ( jω) Ω=ω/T (TX (e jω )) 40
Continuous-Time Processing of Discrete-Time Y c ( jω) = H c ( jω)x c ( jω) Y (e jω ) = 1 T Y c ( jω) Ω=ω/T Y (e jω ) = 1 T H c ( jω) Ω=ω/T X c ( jω) Ω=ω/T = 1 T H c ( jω) Ω=ω/T (TX (e jω )) ω < π H(e jω ) 41
Example 42
Example: Non-integer Delay! What is the time domain operation when Δ is noninteger? I.e Δ=1/2 δ[n] 1 δ[n n d ] e jωn d 43
Example: Non-integer Delay! What is the time domain operation when Δ is noninteger? I.e Δ=1/2 delay of ΔT in continuous time 44
Example: Non-integer Delay! What is the time domain operation when Δ is noninteger? I.e Δ=1/2 delay of ΔT in continuous time 45
Example: Non-integer Delay 46
Example: Non-integer Delay 47
Example: Non-integer Delay 48
Example: Non-integer Delay! The block diagram is for interpretation/analysis only y c (t) = x c (t TΔ) 49
Example: Non-integer Delay! The block diagram is for interpretation/analysis only y c (t) = x c (t TΔ) y[n] = y c (nt ) = x c (nt TΔ) = x[k] sinc k k t kt TΔ T ( ) = x[k] sinc n k Δ t=nt 50
Example: Non-integer Delay! The block diagram is for interpretation/analysis only y c (t) = x c (t TΔ) y[n] = y c (nt ) = x c (nt TΔ) = x[k] sinc k k t kt TΔ T ( ) = x[k] sinc n k Δ t=nt 51
Example: Non-integer Delay! My delay system has an impulse response of a sinc with a continuous time delay 52
Example: Non-integer Delay! My delay system has an impulse response of a sinc with a continuous time delay 53
Example: Non-integer Delay! My delay system has an impulse response of a sinc with a continuous time delay 54
Example: Non-integer Delay! My delay system has an impulse response of a sinc with a continuous time delay 55
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 56
Admin! HW 3 due Sunday 57