Homework: 4.50 & 4.51 of the attachment Tutorial Problems: 7.41, 7.44, 7.47, Signals & Systems Sampling P1

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1 Homework: 4.50 & 4.51 of the attachment Tutorial Problems: 7.41, 7.44, 7.47, 7.49 Signals & Systems Sampling P1

2 Undersampling & Aliasing Undersampling: insufficient sampling frequency ω s < 2ω M Perfect reconstruction is impossible with undersampling. Aliasing: distortion due to undersampling Signals & Systems Sampling P2

3 Aliasing: Example Signals & Systems Sampling P3

4 Signals & Systems Sampling P4

5 Aliasing in Movies Wheel s rotation in movies Signals & Systems Sampling P5

6 Process Continuous-Time Signals Discretely People would like to process continuous-time signal in discrete-time (digital) domain Signals & Systems Sampling P6

7 Block Diagram It is much easier to design DT system. What s the relation between H C and H D? Signals & Systems Sampling P7

8 Discretization: C/D Conversion Mathematical Interpretation (Fourier Transform) x p (t) = + n= x c (t) X c (jω) x c (nt )δ(t nt ) X p (jω) = 1 T + k= x d [n] = x c (nt ) X d (e jω ) = X p (jω/t ) X c (j(ω kω s )) Signals & Systems Sampling P8

9 DT Processing and Conversion Mathematical Interpretation (Fourier Transform) y d [n] = x d [n] h D [n] Y d (e jω ) = X d (e jω )H D (e jω ) y p (t) = y d [n]δ(t nt ) Y p (jω) = Y d (e jωt ) n= y(t) = y p (t) h LP (t) Y (jω) = Y p (jω)h LP (jω) Signals & Systems Sampling P9

10 Input vs. Output Y (jω) = Y p (jω)h LP (jω) = Y d (e jωt )H LP (jω) = X d (e jωt )H D (e jωt )H LP (jω) = X p (jω)h D (e jωt )H LP (jω) [ 1 + ] = X c (j(ω kω s )) H D (e jωt )H LP (jω) T k= = X c (jω)h D (e jωt ) = X c (jω) H D (e jωt ) (1) where H D (e jωt ) = { HD (e jωt ) ω < ω s /2 0 otherwise (2) It is equivalent to a continuous-time LTI system H C (jω) = H D (e jωt ) H D (e jωt ) is a periodic extension of H D (e jωt ) with period ω s = 2π/T Signals & Systems Sampling P10

11 System Design How can we design a CT LTI system with frequency response H C via DT LTI system? Step 1: Sampling frequency ω s or 2π/T should be larger than Nyquist rate Step 2: HD (e jωt ) = H C (jω) Step 3: Frequency response of DT LTI system H D (e jωt ) = H k= D (e j(ω kωs )T ) or H D (e jω ) = H k= D (e j(ω kωs T ) ) = k= H C (j ω 2kπ T ) Signals & Systems Sampling P11

12 System Design Example How to implement an ideal CT lowpass filter? Signals & Systems Sampling P12

13 Sampling on Discrete-Time Signals In digital signal processing, we may sample the discrete-time signals Example: image, audio, video compression Its mathematical model is give below: Signals & Systems Sampling P13

14 Frequency Analysis P(e jω ) = 2π N X p (e jω ) = 1 2π k= δ(ω kω s ) where ω s = 2π/N P(e jθ )X (e j(ω θ) )dθ = 1 N 1 X (e j(ω kωs ) ) 2π N k=0 Signals & Systems Sampling P14

15 Reconstruction Perfect reconstruction is applicable when ω s > 2ω M N < π ω M Aliasing occurs when ω s < 2ω M Signals & Systems Sampling P15

16 Decimation After sampling, there will be a great amount of redundancy Decimation: discrete-time sampling + remove zeros x b [n] = x p [nn] = x[nn] Signals & Systems Sampling P16

17 Frequency Analysis X b (e jω ) = x b [n]e jωn = x p[n]e jωn/n = X p(e jω/n ) n= n= = 1 N 1 X (e jω/n kωs ) N k=0 Signals & Systems Sampling P17

18 Anything Else? Downsampling: to reduce the sampling frequency (decimation) Upsampling: to generate a DT signal with higher sampling frequency As long as Nyquist rate is satisfied, the transform between low-sampling-frequency version and high-sampling-frequency versions is lossless. Signals & Systems Sampling P18

19 Downsampling Downsampling: a general procedure to reduce the sampling frequency When do we need downsampling? Signals & Systems Sampling P19

20 Downsampling Example (1/2) Suppose we have a clip of voice, x(t), with bandwidth =19.9kHz It can be converted to DT signal with sampling frequency 40kHz, denoted as x 1 [n] Based on x 1 [n], if we want to save the voice information within 9.9kHz into another DT signal, what can we do? Signals & Systems Sampling P20

21 Downsampling Example (2/2) Signals & Systems Sampling P21

22 Upsampling Upsampling: a procedure to generate a sequence with higher sampling frequency Superpose the following two digital sound clips Audio clip 1: Bandwidth= 19.9kHz, sampled at 40kHz Audio clip 2: Bandwidth= 9.9kHz, sampled at 20kHz Double the sampling frequency of audio clip 2 (40kHz) How to do upsampling in discrete-time domain? Signals & Systems Sampling P22

23 Time expansion (Insert zeros): x p [n] = x b(k) [n] X p (e jω ) = X b (e jkω ) Low-pass filtering: Signals & Systems Sampling P23