0195136047.pdf

14.1 SIGNALS AND SPECTRAL ANALYSIS 639

Figure 14.1.5 shows a block diagram in which an arbitrary linear network characterized by its
ac transfer functionH(jω) has an input signalx(t) yielding an output signaly(t). Note thatH(jω)
is represented here in terms of theamplitude ratioandphase shiftas a function of frequencyf
given by|H(f)|=|H(jω)|andθ(f)= H(jω), respectively, whereωis related tofthrough
the relationω= 2 πf.
Ifx(t) contains a sinusoidal component of magnitudeA 1 and phaseφ 1 at frequencyf 1 , the
corresponding output component of the linear network will have amplitude|H(f 1 )|A 1 and phase
φ 1 +θ(f 1 ). If, on the other hand, the input should consist of several sinusoids given by

x(t)=

∑

n

Ancos( 2 πfnt+θn) (14.1.16)

By superposition, the steady-state response at the output will be

y(t)=

∑

n

|H(fn)|Ancos[ 2 πfnt+φn+θ(fn)] (14.1.17)

By lettingfn=nf 1 withn=0, 1, 2,... ,periodic steady-state responsedue to a periodic signal
can be obtained.
Whether periodic or nonperiodic, the output waveform signal is said to beundistortedif the
output is of the form
y(t)=Kx(t−td) (14.1.18)

That is to say, the output has the same shape as the inputscaledby a factorKanddelayedin time
bytd. Fordistortionless transmissionthrough a network it follows then that

|H(f)|=K and θ(f)=−360°(tdf) (14.1.19)

which must hold for all frequencies in the input signalx(t). Thus, a distortionless network will
have a constant amplitude ratio and a negative linear phase shift over the frequency range in
question.
When alow-pass signalhaving a bandwidthWis applied to alow-pass filter(see Section
3.4) with bandwidthB, essentially distortionless output is obtained whenB≥W. Figure 14.1.6
illustrates the frequency-domain interpretation of distortionless transmission.
The preceding observation is of practical interest because many information-bearing wave-
forms are low-pass signals, and transmission cables often behave like low-pass filters. Also notice
that unwanted components atf>Wcontained in a low-pass signal can be eliminated by low-pass
filtering without distorting the filtered output waveform.
If|H(f)| = K, one of the conditions given by Equation (14.1.19) for distortionless
transmission is not satisfied. Then the output suffers fromamplitudeorfrequency distortion,
i.e., the amplitudes of different frequency components are selectively increased or decreased.
Ifθ(f)does not satisfy the condition given in Equation (14.1.19), then the output suffers from
phaseordelay distortion, i.e., different frequency components are delayed by different amounts
of time. Both types oflinear distortionusually occur together.
The linear distortion occurring in signal transmission can often be corrected or reduced by
using anequalizernetwork. The concept is illustrated in Figure 14.1.7, in which an equalizer is
connected at the output of the transmission medium, such that

|H(f)|

∣

∣Heq(f )

∣

∣=K and θ(f)+θeq(f )=−360°(tdf) (14.1.20)

Linear network

|H(f)| ∠θ(f)

x(t) y(t) Figure 14.1.5Linear network with input and output signals.