Chapter 5 Digital Processing of Continuous-Time Signals
Chapter 5 Digital Processing of Continuous-Time Signals
85.1 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 SIgma
§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
85.1 Digital Processing of Continuous-Time Signals Conversion of a continuous-time signal into digital form is carried out by an analog-to-digital (a/D) converter The reverse operation of converting a digital signal into a continuous time signal is performed by a digital-to-analog dla)converter
§5.1 Digital Processing of Continuous-Time Signals • Conversion of a continuous-time signal into digital form is carried out by an analog-to-digital (A/D) converter • The reverse operation of converting a digital signal into a continuoustime signal is performed by a digital-to-analog (D/A) converter
85.1 Digital Processing of Continuous-Time Signals .. Since the ad conversion takes a finite amount of time, a sample-and-hold(s/h) circuit is used to ensure that the analog signal at the input of the a/D converter remains constant in amplitude until the conversion is complete to minimize the error in its representation
§5.1 Digital Processing of Continuous-Time Signals • Since the A/D conversion takes a finite amount of time, a sample-and-hold (S/H) circuit is used to ensure that the analog signal at the input of the A/D converter remains constant in amplitude until the conversion is complete to minimize the error in its representation
85.1 Digital Processing of Continuous-Time Signals To prevent aliasing, an analog anti- aliasing filter is employed before the S/H circuit To smooth the output signal of the D/A converter which has a staircase-like waveform, an analog reconstruction filter is used
§5.1 Digital Processing of Continuous-Time Signals • To prevent aliasing, an analog antialiasing filter is employed before the S/H circuit • To smooth the output signal of the D/A converter, which has a staircase-like waveform, an analog reconstruction filter is used
85.1 Digital Processing of Continuous-Time Signals Complete block-diagram Anti- aliasing S/H HA/DH DSP HD/AH Reconstruction 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 Ir 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 Antialiasing filter S/H A/D D/A Reconstruction DSP filter Complete block-diagram
§52 Sampling of Continuous-time Signals As indicated earlier discrete-time signals in many applications are generated by sampling continuous-time Signals We have seen earlier that identical discrete-time signals may result from the sampling of more than one distinct continuous-time function
§5.2 Sampling of Continuous-time Signals • As indicated earlier, discrete-time signals in many applications are generated by sampling continuous-time signals • We have seen earlier that identical discrete-time signals may result from the sampling of more than one distinct continuous-time function
§52 Sampling of Continuous-time Signals In fact. there exists an infinite number of continuous-time signals, which when sampled lead to the same discrete-time signal However. under certain conditions it is possible to relate a unique continuous time signal to a given discrete-time signal
§5.2 Sampling of Continuous-time Signals • In fact, there exists an infinite number of continuous-time signals, which when sampled lead to the same discrete-time signal • However, under certain conditions, it is possible to relate a unique continuoustime signal to a given discrete-time signal
§52 Sampling of Continuous-time Signals If these conditions hold then it is possible to recover the original continuous-time signal from its sampled values We next develop this correspondence and the associated conditions
§5.2 Sampling of Continuous-time Signals • If these conditions hold, then it is possible to recover the original continuous-time signal from its sampled values • We next develop this correspondence and the associated conditions
§52 Sampling of Continuous-time Signals Let ga(t be a continuous-time signal that is sampled uniformly at t=nT, generating the sequence gn where gIn]=ga(nT), 0<n<0 with t being the sampling period The reciprocal of T is called the sampling frequency FT i. e FT=1/T
§5.2 Sampling of Continuous-time Signals • Let ga (t) be a continuous-time signal that is sampled uniformly at t = nT, generating the sequence g[n] where g[n] = ga (nT), - < n < with T being the sampling period • The reciprocal of T is called the sampling frequency FT , i.e., FT =1/T