1. SIGNAL DISTORTION IN TRANSMISSION
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• Distortionless Transmission
• Linear Distortion
• Equalization
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2. 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 constant
amplitude response and negative linear phase shift, so
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4. Major types of distortion:
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1. Amplitude distortion, which occurs when
2. Delay distortion, which occurs when
3. Nonlinear distortion, which occurs when the system
includes nonlinear elements
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5. Linear Distortion
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Linear distortion includes any amplitude or delay distortion associated
with a linear transmission system.
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6. 6
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Shifting each component by
one-fourth cycle, θ = –90°.
The peak excursions of the phase-shifted signal are substantially greater (by
about 50 percent) than those of the input test signal
This is not due to amplitude response, it is because the components of the
distorted signal all attain maximum or minimum values at the same time, which 7
was not true of the input.

8. Equalization
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Linear distortion—both amplitude and delay—is theoretically curable
through the use of equalization networks.
Since the overall transfer function is H(f) = HC(f)Heq(f) the final output will
be distortionless if HC(f)Heq(f) = Ke-jωtd, where K and td are more or less
arbitrary constants. Therefore, we require that
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wherever X(f) ≠ 0

9. FILTERS AND FILTERING
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• Ideal Filters
• Bandlimiting and Timelimiting
• Real Filters
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10. 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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11. an ideal lowpass filter (LPF) is defined by
an ideal highpass filter (HPF) has
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Ideal band-rejection or notch filters provide distortionless transmission over all
frequencies except some stopband, say
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12. an ideal LPF whose transfer function, shown in Fig. above,
can be written as
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H.W. Explain why the LPF is noncausal
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13. Bandlimiting and Timelimiting
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A strictly bandlimited signal cannot be timelimited.
Conversely, by duality, a strictly timelimited signal cannot be
bandlimited.
Perfect bandlimiting and timelimiting are
mutually incompatible.
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A strictly timelimited signal is not strictly bandlimited, its
spectrum may be negligibly small above some upper
frequency limit W.
A strictly bandlimited signal may be negligibly small outside a
certain time interval t1 ≤ t ≥ t2. Therefore, we will often
assume that signals are essentially both bandlimited and
timelimited for most practical purposes.
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15. Real Filters
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16. nth-order Butterworth LPF
The transfer function with has the form
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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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19. EXAMPLE Second-order LPF
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20. From Table 3.4–1 with p = jf/B , we want
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The required relationship between R, L, and C that satisfies the equation
can be found by setting
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21. QUADRATURE FILTERS AND HILBERT
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TRANSFORMS
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22. A quadrature filter is an allpass network that merely shifts
the phase of positive frequency components by -90° and
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negative 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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23. The corresponding impulse response is
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applying duality to
which yields
so
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24. Now let an arbitrary signal x(t) be the input to a quadrature filter.
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defined as the Hilbert transform of x(t)
denoted by
the spectrum of
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25. Assume that the signal x(t) is real.
1. A signal x(t) and its Hilbert transform have the same
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amplitude spectrum. In addition, the energy or power in a
signal and its Hilbert transform are also equal.
2. If is the Hilbert transform of x(t), then –x(t) is the
Hilbert transform of
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26. 3. A signal x(t) and its Hilbert transform are
orthogonal.
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27. EXAMPLE Hilbert Transform of a Cosine Signal
If the input is
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