Signals and Systems - Electrical Engineering

710 CHAPTER 12: Applications of Discrete-Time Signals and Systems

indicated before, digital communications began with the introduction of pulse code modulation

(PCM). Telephony and radio using baseband and band-pass signals have converged into wireless

communications. Many of the principles of analog communications have remained, but its

implementation has changed from analog to digital with slightly different objectives. Efficient use

of the radio spectrum and efficient processing have become the objectives of modern wireless com-

munication systems such as spread spectrum and orthogonal frequency-division multiplexing, which

we introduce here.

12.2 Application to Digital Signal Processing

In many applications, such as speech processing or acoustics, one would like to digitally process

analog signals. In practice, this is possible by converting the analog signals into binary signals using

an analog-to-digital converter (ADC), and if the output is desired in analog form a digital-to-analog

converter (DAC) is used to convert the binary signal into a continuous-time signal. Ideally, if no

quantization is considered and if the discrete-time signal is converted into an analog signal by sinc

interpolation the system can be visualized as in Figure 12.1.

Viewing the whole system as a black box with an analog signalx(t)as input, and giving as output

also an analog signaly(t), the processing can be seen as a continuous-time system with a transfer

functionG(s). Under the assumption of no quantization, the discrete-time signalx[n] is obtained by

samplingx(t)using a sampling period determined by the Nyquist sampling condition. Likewise, con-

sidering the transformation of a discrete-time (or sampled signal)y[n] into a continuous-time signal

y(t)by means of the sinc interpolation, the ideal DAC is an analog low-pass filter that interpolates

the discrete-time samples to obtain an analog signal. Finally, the discrete-time signalx[n] is pro-

cessed by a discrete-time system with transfer functionH(z), which depends on the desired transfer

functionG(s).

Thus, one can process discrete- or continuous-time signals using discrete systems. A great deal of the

computational cost of this processing is due to the convolution sum used to obtain the output of

the discrete system. That is where the significance of the FFT algorithm lies. Although the DFT allows

us to simplify the convolution to a multiplication, it is the FFT that as an algorithm provides a very

efficient implementation of this process. In the next section, we will introduce you to the FFT and

provide some of the basics of this algorithm for you to understand its efficiency.

FIGURE 12.1

Discrete processing of analog signals

using an ideal ADC and DAC.G(s)is

the transfer function of the overall

system, whileH(z)is the transfer

function of the discrete-time system.

Discrete system

Equivalent analog system

G(s)

x(t)

x[n] y[n]

H(z) Hr(jΩ) y(t)

Ideal ADC Ideal DAC