Signal Processing and Linear Systems-B.P.Lathi copy

D t t 0 t 2 t t 452 t his r esult applies only

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Unformatted text preview: )1 is t he amplitude response, and LH(w) is t he phase response of the system. T he plots o f IH(w)1 a nd LH(w) as functions of w show a t a glance how the system modifies t he amplitudes a nd phases of various sinusoidal inputs. For this reason, H (w) is c alled the f requency r esponse of the system. During transmission through the system, some frequency components may be boosted in amplitude, while others may be attenuated. T he relative phases of the various components also change. In general, t he o utput waveform will be different from the input waveform. Distortionless Transmission I n several applications, such as signal amplification or message signal transmission over a communication channel, we require t hat t he o utput waveform be a replica of t he i nput waveform. In such cases we need to minimize the distortion caused by t he amplifier or the communication channel. I t is, therefore, of practical interest to determine the characteristics of a system t hat allows a signal t o pass without distortion ( distortionless t ransmission). Transmission is said t o b e distortionless if the input and the o utput have identical waveshapes within a multiplicative constant. A delayed o utput t hat r etains the input waveform is also considered distortionless. Thus, in distortionless transmission, t he i nput f (t) a nd the o utput y(t) satisfy the condition T his is t he transfer function required for d istortion less transmission. From this equation i t follows t hat (4.58a) IH(w)1 = k LH(w) = - wtd This result shows t hat for d istortion less transmission, t he a mplitude response IH (w)1 must be a constant, and the phase response LH(w) m ust be a linear function of w with slope - td, where td is t he delay of the o utput w ith respect t o i nput (Fig. 4.26). Intuitive Explanation o f t he Distortionless Transmission Conditions I t is i nstructive t o derive the conditions for distortionless transmission heuristically. Once again, imagine f (t) t o b e composed of various sinusoids (its spectral components), which are being passed through a distortionless system. For the distortionless case, the o utput signal is t he i nput signal multiplied by k a nd delayed by td. To synthesize such a signal, we need exactly t he same components as those of f (t), w ith each component multiplied by k a nd delayed by td. Thus, t he system transfer function H (w) s hould be such t hat each sinusoidal component suffers the same attenuation k a nd each component undergoes t he same time delay of td seconds. The first condition requires t hat IH(w)1 = k We have seen in our discussion on p. 258 t hat t o achieve t he same time delay td for every frequency component requires a linear phase delay wtd (Fig. 4.20). Therefore LH(w) = - wtd This equation shows t hat t he time delay resulting from signal transmission through a system is t he negative of the slope of the system phase response LH (w ); t hat is, d y(t) = k f(t - td) (4.57) (4.58b) td(W) = - -LH(w) dw (4.59) 270 4 C ontinuous-Time Signal Analysis: T he F ourier Transform I f t he slope o f LH (w) is c onstant ( that is, if LH (w) is linear w ith w), all t he components a r...
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