Chapter 5 Digital Processing of Continuous-Time Signals
Chapter 5 Digital Processing of Continuous-Time Signals
5.1 Digital Processing of Continuous-Time Signals Digital processing of a continuous-time signal involves the following basic steps: (1)Conversion of the continuous-time signal into a discrete-time signal, (2)Processing of the discrete-time signal, (3)Conversion of the processed discrete- time signal back into a continuous-time signal
§5.1 Digital Processing of Continuous-Time Signals • Digital processing of a continuous-time signal involves the following basic steps: (1) Conversion of the continuous-time signal into a discrete-time signal, (2) Processing of the discrete-time signal, (3) Conversion of the processed discretetime signal back into a continuous-time signal
5.1 Digital Processing of Continuous-Time Signals Complete block-diagram Anti- aliasing S/H A/D DSP D/A Reconstruction filter filter Since both the anti-aliasing filter and the reconstruction filter are analog lowpass filters, we review first the theory behind the design of such filters Also,the most widely used IIR digital filter design method is based on the conversion of an analog lowpass prototype
§5.1 Digital Processing of Continuous-Time Signals • Since both the anti-aliasing filter and the reconstruction filter are analog lowpass filters, we review first the theory behind the design of such filters • Also, the most widely used IIR digital filter design method is based on the conversion of an analog lowpass prototype Anti- aliasing filter S/H A/D D/A Reconstruction DSP filter Complete block-diagram
§5.2 Sampling of Continuous-time Signals The frequency-domain representation of ga(t)is given by its continuos-time Fourier transform (CTFT): Ga(j)=ga(t)e idi The frequency-domain representation of gn is given by its discrete-time Fourier transform (DTFT): G(eo)=∑m-og[neon
§5.2 Sampling of Continuous-time Signals • The frequency-domain representation of ga(t) is given by its continuos-time Fourier transform (CTFT): G j g t e dt j t a ∫ a ∞ −∞ − Ω ( Ω) = ( ) ∑∞ =−∞ ω − ω = n j j n G(e ) g[n]e • The frequency-domain representation of g[n] is given by its discrete-time Fourier transform (DTFT):
5.3 Effect of Sampling in the Frequency Domain To establish the relation between Ga(js) and G(ei),we treat the sampling operation mathematically as a multiplication of ga(t)by a periodic impulse train p(t): 00 p(t)=∑δ(t-nT) n=-00 p(1)
§5.3 Effect of Sampling in the Frequency Domain • To establish the relation between Ga(jΩ) and G(ejω) , we treat the sampling operation mathematically as a multiplication of ga(t) by a periodic impulse train p(t): = ∑δ − ∞ n=−∞ p(t) (t nT) g (t) × a g (t) p p(t)
5.3 Effect of Sampling in the Frequency Domain gp(t)is a continuous-time signal consisting of a train of uniformly spaced impulses with the impulse at t=nT weighted by the sampled value ga(nT)of ga(t)at that instant t=nT &a(0 8p) 8a() 0 2T-T0T2T、 8a(4D
§5.3 Effect of Sampling in the Frequency Domain • gp(t) is a continuous-time signal consisting of a train of uniformly spaced impulses with the impulse at t = nT weighted by the sampled value ga(nT) of ga(t) at that instant t=nT
5.3 Effect of Sampling in the Frequency Domain The impulse train gp(t)can be expressed as 0-豆n From the frequency-shifting property of the CTFT,the CTFT of eirkiga(t)is given by Ga(j(-kT))
§5.3 Effect of Sampling in the Frequency Domain • The impulse train gp(t) can be expressed as ( ) ( ) 1 g t e g t a k j kt T p T ⋅ = ∑ ∞ =−∞ Ω • From the frequency-shifting property of the CTFT, the CTFT of ejΩTktga(t) is given by Ga(j(Ω - kΩT))
5.3 Effect of Sampling in the Frequency Domain Hence,the CTFT of gp(t)is given by Gm(2)=,∑GaU2-k2r》 Therefore,Go(js)is a periodic function of consisting of a sum of shifted and scaled replicas of Ga(j),shifted by integer multiples of r and scaled by 1/T
§5.3 Effect of Sampling in the Frequency Domain • Hence, the CTFT of gp(t) is given by ∑ ( ) ∞ =−∞ Ω = Ω − Ω k a T T p G ( j ) G j( k ) 1 • Therefore, Gp(jΩ) is a periodic function of Ω consisting of a sum of shifted and scaled replicas of Ga(jΩ) , shifted by integer multiples of ΩT and scaled by 1/T
5.3 Effect of Sampling in the Frequency Domain The term on the RHS of the previous equation for k 0 is the baseband portion of G(j),and each of the remaining terms are the frequency translated portions of Gp(j) The frequency range s2≤ 2 2 is called the baseband or Nyquist band
§5.3 Effect of Sampling in the Frequency Domain • The term on the RHS of the previous equation for k = 0 is the baseband portion of Gp(jΩ) , and each of the remaining terms are the frequency translated portions of Gp(jΩ) • The frequency range 2 2 T ΩT ≤ Ω − Ω ≤ is called the baseband or Nyquist band
5.3 Effect of Sampling in the Frequency Domain Assume ga(t)is a band-limited signal with a CTFT Ga(j)as shown below G.(j) 0 The spectrum P(j)of p(t)having a sampling period T=2m/r is indicated below P(jQ) 1 0 r 227 391
§5.3 Effect of Sampling in the Frequency Domain • Assume ga(t) is a band-limited signal with a CTFT Ga(jΩ) as shown below • The spectrum P(jΩ) of p(t) having a sampling period T=2π/ΩT is indicated below