0.1 A telecommunication system  (Page 6/15)

This is discussed in detail in [link] . Thus, the convolution of a linear filter can readily be viewedin the frequency (Fourier) domain as a point-by-point multiplication. For instance, an ideal lowpass filter (LPF) passes all frequencies below $f_l$ (which is called the cutoff frequency).This is commonly plotted in a curve called the frequency response of the filter, which describes the action of the filter. Formally, the frequency response can be calculated as the Fourier transform of the impulseresponse of the filter. If this filter is applied to a signal $w\left(t\right)$ , then all energy above $f_l$ is removed from $w\left(t\right)$ . [link] shows this pictorially. If $w\left(t\right)$ has the magnitude spectrum shown in part (a), and the frequency response of the lowpass filter withcutoff frequency $f_l$ is as shown in part (b), then the magnitude spectrum of the output appearsin part (c).

The problem of how to design and implement such filters is considered in detail in Chapter [link] .

Analog downconversion

Because transmitters typically modulate the message signal with a high frequency carrier,the receiver must somehow remove the carrier from the message that it carries.One way is to multiply the received signal by a cosine wave of the same frequency (and the same phase)as was used at the transmitter. This creates a (scaled) copy of the original signalcentered at zero frequency, plus some other high frequency replicas.A lowpass filter can then remove everything but the scaled copy of the original message. This is how the boxlabelled “frequency translator” in [link] is typically implemented.

To see this procedure in detail, suppose that $s\left(t\right)=w\left(t\right)\cos \left(2\pi f_{0}t\right)$ arrives at the receiver, which multiplies $s\left(t\right)$ by another cosine wave of exactly the same frequency and phase to get the demodulated signal

Using the trigonometric identity [link] , namely,

we find that this can be rewritten as

The spectrum of the demodulated signal can be calculated

by linearity. Now the frequency shifting property [link] can be applied to show that

Thus, the spectrum of this downconverted received signal has the original baseband component (scaled to 50%)and two matching pieces (each scaled to 25%) centered around twice the carrier frequency $f_0$ and twice its negative. A lowpass filter can now be used to extract $W\left(f\right)$ , and hence to recover the original message $w\left(t\right)$ .