SIGNAL DISTORTION IN TRANSMISSION

• Distortionless Transmission• Linear Distortion• Equalization

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Distortionless Transmission

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Distortionless transmission means that the output signal has the same“shape” as the input.

The output is undistorted if it differs from the input only by a multiplying constant and a finite time delay

Analytically, we have distortionless transmission if

where K and td are constants.

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Now by definition of transfer function, Y(f) = H(f)X(f) , so

A system giving distortionless transmission must have constantamplitude response and negative linear phase shift, so

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Major types of distortion:

1. Amplitude distortion, which occurs when

2. Delay distortion, which occurs when

3. Nonlinear distortion, which occurs when the systemincludes nonlinear elements

Linear Distortion

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Linear distortion includes any amplitude or delay distortion associatedwith a linear transmission system.

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Shifting each component by one-fourth cycle, θ = –90°.

The peak excursions of the phase-shifted signal are substantially greater (byabout 50 percent) than those of the input test signal

This is not due to amplitude response, it is because the components of thedistorted signal all attain maximum or minimum values at the same time, whichwas not true of the input.

Equalization

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Linear distortion—both amplitude and delay—is theoretically curablethrough the use of equalization networks.

Since the overall transfer function is H(f) = HC(f)Heq(f) the final output willbe distortionless if HC(f)Heq(f) = Ke-jωtd, where K and td are more or lessarbitrary constants. Therefore, we require that

wherever X(f) ≠ 0

FILTERS AND FILTERING

• Ideal Filters• Bandlimiting and Timelimiting• Real Filters

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Ideal Filters

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The transfer function of an ideal bandpass filter (BPF) is

The filter’s bandwidth is

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an ideal lowpass filter (LPF) is defined by

an ideal highpass filter (HPF) has

Ideal band-rejection or notch filters provide distortionless transmission over all frequencies except some stopband, say

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an ideal LPF whose transfer function, shown in Fig. above, can be written as

H.W. Explain why the LPF is noncausal

Bandlimiting and Timelimiting

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A strictly bandlimited signal cannot be timelimited.Conversely, by duality, a strictly timelimited signal cannot bebandlimited.

Perfect bandlimiting and timelimiting are mutually incompatible.

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A strictly timelimited signal is not strictly bandlimited, itsspectrum may be negligibly small above some upperfrequency limit W.

A strictly bandlimited signal may be negligibly small outside acertain time interval t1 ≤ t ≥ t2. Therefore, we will oftenassume that signals are essentially both bandlimited andtimelimited for most practical purposes.

Real Filters

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nth-order Butterworth LPF

The transfer function with has the form

where B equals the 3 dB bandwidth and Pn(jf/B) is a complex polynomial

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normalized variable p= jf/B

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EXAMPLE Second-order LPF

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From Table 3.4–1 with p = jf/B , we want

The required relationship between R, L, and C that satisfies the equation can be found by setting

QUADRATURE FILTERS AND HILBERT TRANSFORMS

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A quadrature filter is an allpass network that merely shiftsthe phase of positive frequency components by -90° andnegative frequency components by +90°.

Since a ±90° phase shift is equivalent to multiplying by ,

The transfer function can be written in terms of the signum function as

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The corresponding impulse response is

applying duality to

which yields

so

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Now let an arbitrary signal x(t) be the input to a quadrature filter.

defined as the Hilbert transform of x(t)denoted by

the spectrum of

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Assume that the signal x(t) is real.

1. A signal x(t) and its Hilbert transform have the sameamplitude spectrum. In addition, the energy or power in asignal and its Hilbert transform are also equal.

2. If is the Hilbert transform of x(t), then –x(t) is theHilbert transform of

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3. A signal x(t) and its Hilbert transform areorthogonal.

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EXAMPLE Hilbert Transform of a Cosine Signal

If the input is