10.6 Discrete time processing of continuous time signals  (Page 2/3)

Finally, we will discuss the digital to analog converter, often denoted by DAC or D/A. Since continuous time filters have continuous time inputs and continuous time outputs, we must construct a continuous time signal from our filtered discrete time signal. Assuming that we have sampled a bandlimited at a sufficiently high rate, in the ideal case this would be done using perfect reconstruction through the Whittaker-Shannon interpolation formula. However, there are, once again, practical issues that prevent this from happening that will be discussed later.

Discrete time filter

With some initial discussion of the process illustrated in [link] complete, the relationship between the continuous time, linear time invariant filter $H_1$ and the discrete time, linear time invariant filter $H_2$ can be explored. We will assume the use of ideal, infinite precision ADCs and DACs that perform sampling and perfect reconstruction, respectively, using a sampling rate ${\omega }_s=2\pi /T_s\ge 2B$ where the input signal $x$ is bandlimited to $(-B,B)$ . Note that these arguments fail if this condition is not met and aliasing occurs. In that case, preapplication of an anti-aliasing filter is necessary for these arguments to hold.

Recall that we have already calculated the spectrum $X_s$ of the samples $x_s$ given an input $x$ with spectrum $X$ as

Likewise, the spectrum $Y_s$ of the samples $y_s$ given an output $y$ with spectrum $Y$ is

From the knowledge that $y_s={\left(H_{1}x\right)}_s=H_2\left(x_s\right)$ , it follows that

Because $X$ is bandlimited to $(-\pi /T_s,\pi /T_s)$ , we may conclude that

More simply stated, $H_2$ is $2\pi$ periodic and $H_2\left(\omega \right)=H_1(\omega /T_s)$ for $\omega \in [-\pi ,\pi )$ .

Given a specific continuous time, linear time invariant filter $H_1$ , the above equation solves the system design problem provided we know how to implement $H_2$ . The filter $H_2$ must be chosen such that it has a frequency response where each period has the same shape as the frequency response of $H_1$ on $(-\pi /T_s,\pi /T_s)$ . This is illustrated in the frequency responses shown in [link] .

We might also want to consider the system analysis problem in which a specific discrete time, linear time invariant filter $H_2$ is given, and we wish to describe the filter $H_1$ . There are many such filters, but we can describe their frequency responses on $(-\pi /T_s,\pi /T_s)$ using the above equation. Isolating one period of $H_2\left(\omega \right)$ yields the conclusion that $H_1\left(\omega \right)=H_2\left(\omega T_s\right)$ for $\omega \in (-\pi /T_s,\pi /T_s)$ . Because $x$ was assumed to be bandlimited to $(-\pi /T,\pi /T)$ , the value of the frequency response elsewhere is irrelevant.

Practical considerations

As mentioned before, there are several practical considerations that need to be addressed at each stage of the process shown in [link] . Some of these will be briefly addressed here, and a more complete model of how discrete time processing of continuous time signals appears in [link] .