ESE 531: Digital Signal Processing Lec 8: February 7th, 2017 Sampling and Reconstruction
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??) 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 s (e jω ) ω = ΩT Ω s 2 T = π 7
Frequency Domain Analysis X s (e jω ) ω = ΩT 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 13
Anti-Aliasing Filter 14
MRI aliasing example 15
MRI anti-aliasing example 16
MRI anti-aliasing example 17
Reconstruction in Frequency Domain Different T? 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 19
Discrete-Time Processing of Continuous Time x[n] y[n] T T X c ( j(ω kω s )) X (e jω ) = 1 T k= X c ω j T 2πk T 20
Frequency Domain Analysis X s (e jω ) ω = ΩT Ω s 2 T = π 21
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 n= y r (t) = y[n] sin[π (t nt ) / T ] π (t nt ) / T 22
Reconstruction in Frequency Domain 23
Reconstruction in Time Domain * = The sum of sincs gives x r (t) # unique signal that is bandlimited by sampling bandwidth 24
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 25
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] is LTI, H(e jω ) exists " Is the whole system from x c (t)#y c (t) LTI? 26
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 27
Example! Consider the following system! 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 28
Impulse Invariance! Want to implement continuous-time system in discrete-time 29
Impulse Invariance! With H c (jω) bandlimited, choose H(e jω ) = H c ( jω / T ), ω < π! With the further requirement that T be chosen such that H c ( jω) = 0, Ω π / T 30
Impulse Invariance! With H c (jω) bandlimited, choose H(e jω ) = H c ( jω / T ), ω < π! With the further requirement that T be chosen such that H c ( jω) = 0, Ω π / T h[n] = Th c (nt ) 31
Continuous-Time Processing of Discrete-Time! Useful to interpret DT systems with no simple interpretation in discrete time n= x c (t) = x[n] sin[π (t nt ) / T ] π (t nt ) / T n= y c (t) = y[n] sin[π (t nt ) / T ] π (t nt ) / T 32
Continuous-Time Processing of Discrete-Time X c ( jω) = TX (e jωt ) Ω < π / T 0 else 33
Continuous-Time Processing of Discrete-Time X c ( jω) = TX (e jωt ) Ω < π / T 0 else Y c ( jω) = H c ( jω)x c ( jω) Also bandlimited 34
Continuous-Time Processing of Discrete-Time Y c ( jω) = H c ( jω)x c ( jω) Also bandlimited Y (e jω ) = 1 T Y c k= ( ) j Ω kω s Ω=ω/T = 1 T Y c ( jω) Ω=ω/T 35
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 )) = H(e jω )X (e jω ) ω < π 36
Example 37
Example: Non-integer Delay! What is the time domain operation when Δ is noninteger? I.e Δ=1/2 38
Reminder: Properties of the DTFT! Time Reversal: x[n] X (e jω ) x[ n] X (e jω ) If x[n] real x[ n] X *(e jω )! Time/Freq Shifting: x[n] X (e jω ) x[n n d ] e jωn d X (e jω ) e jω 0 n x[n] X (e j(ω ω 0 ) ) 39
Example: Non-integer Delay! What is the time domain operation when Δ is noninteger? I.e Δ=1/2 40
Example: Non-integer Delay 41
Example: Non-integer Delay 42
Example: Non-integer Delay! The block diagram is for interpretation/analysis only y c (t) = x c (t TΔ) 43
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 44
Example: Non-integer Delay! My delay system has an impulse response of a sinc with a continuous time delay 45
Example: Non-integer Delay! My delay system has an impulse response of a sinc with a continuous time delay 46
Example: Non-integer Delay! My delay system has an impulse response of a sinc with a continuous time delay 47
Example: Non-integer Delay! My delay system has an impulse response of a sinc with a continuous time delay 48
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 49
Admin! HW 2 due Friday! Ahead of schedule " Watch course calendar online for changes 50