1. Discrete-Time Processing of Continuous-Time Signals Quote of the Day Do not worry about your difficulties in Mathematics. I can assure you mine are still greater. Albert Einstein Content and Figures are from Discrete-Time Signal Processing, 2e by Oppenheim, Shafer, and Buck, ©1999-2000 Prentice Hall Inc.

2. Copyright (C) 2005 Güner Arslan 351M Digital Signal Processing 2 Reconstruction of Bandlimited Signal From Samples Sampling can be viewed as modulating with impulse train If Sampling Theorem is satisfied –The original continuous-time signal can be recovered –By filtering sampled signal with an ideal low-pass filter (LPF) Impulse-train modulated signal Pass through LPF with impulse response h r (t) to reconstruct Ideal reconstruction filter H r (j  ) x[n]x r (t) Convert from sequence to impulse train

3. Copyright (C) 2005 Güner Arslan 351M Digital Signal Processing 3 Ideal Reconstruction Filter Ideal LPC with cut of frequency of  c =/T or f c =2/T

4. Copyright (C) 2005 Güner Arslan 351M Digital Signal Processing 4 Reconstructed Signal sinc function is 1 at t=0 sinc function is 0 at nT

5. Copyright (C) 2005 Güner Arslan 351M Digital Signal Processing 5 Discrete-Time Processing of Continuous-Time Signals Overall system is equivalent to a continuous-time system –Input and output is continuous-time The continuous-time system depends on –Discrete-time system –Sampling rate We’re interested in the equivalent frequency response –First step is the relation between x c (t) and x[n] –Next between y[n] and x[n] –Finally between y r (t) and y[n] D/C x c (t)y r (t) C/D Discrete- Time System

6. Copyright (C) 2005 Güner Arslan 351M Digital Signal Processing 6 Effective Frequency Response Input continuous-time to discrete-time Assume a discrete-time LTI system Output discrete-time to continuous-time Output frequency response Effective Frequency Response

7. Copyright (C) 2005 Güner Arslan 351M Digital Signal Processing 7 Example Ideal low-pass filter implemented as a discrete-time system Continuous-time input signal Sampled continuous- time input signal Apply discrete-time LPF

8. Copyright (C) 2005 Güner Arslan 351M Digital Signal Processing 8 Example Continued Signal after discrete- time LPF is applied Application of reconstruction filter Output continuous- time signal after reconstruction

9. Copyright (C) 2005 Güner Arslan 351M Digital Signal Processing 9 Impulse Invariance Given a continuous-time system H c (j) –how to choose discrete-time system response H(e j ) –so that effective response of discrete-time system H eff (j)=H c (j) Answer: Condition: Given these conditions the discrete-time impulse response can be written in terms of continuous-time impulse response as Resulting system is the impulse-invariant version of the continuous-time system

10. Copyright (C) 2005 Güner Arslan 351M Digital Signal Processing 10 Example: Impulse Invariance Ideal low-pass discrete-time filter by impulse invariance The impulse response of continuous-time system is Obtain discrete-time impulse response via impulse invariance The frequency response of the discrete-time system is