1. SIGNAL DISTORTION IN TRANSMISSION
11/16/2011 11:08 AM
• Distortionless Transmission
• Linear Distortion
• Equalization
1
2. Distortionless Transmission
11/16/2011 11:08 AM
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.
2
3. 11/16/2011 11:08 AM
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
3
4. Major types of distortion:
11/16/2011 11:08 AM
1. Amplitude distortion, which occurs when
2. Delay distortion, which occurs when
3. Nonlinear distortion, which occurs when the system
includes nonlinear elements
4
5. Linear Distortion
11/16/2011 11:08 AM
Linear distortion includes any amplitude or delay distortion associated
with a linear transmission system.
5
7. 11/16/2011 11:08 AM
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
11/16/2011 11:08 AM
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
8
wherever X(f) ≠ 0
9. FILTERS AND FILTERING
11/16/2011 11:08 AM
• Ideal Filters
• Bandlimiting and Timelimiting
• Real Filters
9
10. Ideal Filters
11/16/2011 11:08 AM
The transfer function of an ideal bandpass filter (BPF) is
The filter’s bandwidth is
10
11. an ideal lowpass filter (LPF) is defined by
an ideal highpass filter (HPF) has
11/16/2011 11:08 AM
Ideal band-rejection or notch filters provide distortionless transmission over all
frequencies except some stopband, say
11
12. an ideal LPF whose transfer function, shown in Fig. above,
can be written as
11/16/2011 11:08 AM
H.W. Explain why the LPF is noncausal
12
13. Bandlimiting and Timelimiting
11/16/2011 11:08 AM
A strictly bandlimited signal cannot be timelimited.
Conversely, by duality, a strictly timelimited signal cannot be
bandlimited.
Perfect bandlimiting and timelimiting are
mutually incompatible.
13
14. 11/16/2011 11:08 AM
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.
14
16. nth-order Butterworth LPF
The transfer function with has the form
11/16/2011 11:08 AM
where B equals the 3 dB bandwidth and Pn(jf/B) is a complex polynomial
16
20. From Table 3.4–1 with p = jf/B , we want
11/16/2011 11:08 AM
The required relationship between R, L, and C that satisfies the equation
can be found by setting
20
22. A quadrature filter is an allpass network that merely shifts
the phase of positive frequency components by -90° and
11/16/2011 11:08 AM
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
22
24. Now let an arbitrary signal x(t) be the input to a quadrature filter.
11/16/2011 11:08 AM
defined as the Hilbert transform of x(t)
denoted by
the spectrum of
24
25. Assume that the signal x(t) is real.
1. A signal x(t) and its Hilbert transform have the same
11/16/2011 11:08 AM
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
25
26. 3. A signal x(t) and its Hilbert transform are
orthogonal.
11/16/2011 11:08 AM
26
