DiscreteTime Processing of ContinuousTime Signals Quote of the

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 Copyright (C) 2005 Güner 351 M Digital 2 e. Signal Processing Content and Figures are. Arslan from Discrete-Time Signal Processing, by Oppenheim, Shafer, and Buck, © 1999 -2000 Prentice Hall Inc.

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 hr(t) to reconstruct x[n] Copyright (C) 2005 Güner Arslan Convert from sequence to impulse train Ideal reconstruction filter Hr(j ) 351 M Digital Signal Processing xr(t) 2

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

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

Discrete-Time Processing of Continuous-Time Signals xc(t) Discrete. Time System C/D D/C yr(t) • 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 xc(t) and x[n] – Next between y[n] and x[n] – Finally between yr(t) and y[n] Copyright (C) 2005 Güner Arslan 351 M Digital Signal Processing 5

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 Copyright (C) 2005 Güner Arslan 351 M Digital Signal Processing 6

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

Example Continued Signal after discretetime LPF is applied Application of reconstruction filter Output continuoustime signal after reconstruction Copyright (C) 2005 Güner Arslan 351 M Digital Signal Processing 8

Impulse Invariance • Given a continuous-time system Hc(j ) – how to choose discrete-time system response H(ej ) – so that effective response of discrete-time system Heff(j )=Hc(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 Copyright (C) 2005 Güner Arslan 351 M Digital Signal Processing 9

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 Copyright (C) 2005 Güner Arslan 351 M Digital Signal Processing 10