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)