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Sec. 2–7 Bandlimited Signals and Noise 89
the phase error is less than 12.3° (27%). In engineering practice, this type of error is often
considered to be tolerable. Waveforms with spectral components below 0.50f 0 would be delayed
by approximately 1/(2pf 0 ) s, as shown in Fig. 2–16c. That is, if the cutoff frequency of the filter
was f 0 = 1 kHz, the delay would be 0.2 ms. For wideband signals, the higher-frequency
components would be delayed less than the lower-frequency components.
Distortion of Audio, Video, and Data Signals
A linear time-invariant system will produce amplitude distortion if the amplitude response is
not flat, and it will produce phase distortion (i.e., differential time delays) if the phase response
is not a linear function of frequency.
In audio applications, the human ear is relatively sensitive to amplitude distortion,
but insensitive to phase distortion. This is because a phase error of 15° for an audio filter
at 15 kHz would produce a variation (error) in time delay of about 3 µ sec. Comparing
this error to the duration of a spoken syllable, which is in the range of 0.01 to 0.1 sec, the
time delay error due to poor filter phase response is negligible. However, an amplitude error
of 3 dB would certainly be detectable by the human ear. Thus, in linear distortion
specifications of high-fidelity audio amplifiers, one is interested primarily in nonflat
magnitude frequency response characteristics and is not too concerned about the phase
response characteristics.
In analog video applications, the opposite is true: The phase response becomes the
dominant consideration. This is because the human eye is more sensitive to time delay
errors, which result in smearing of an object’s edges, rather than errors in amplitude
(intensity).
For data signals, a linear filter can cause a data pulse in one time slot to smear into
adjacent time slots causing intersymbol interference (ISI). Filter design for minimum ISI is
discussed in Sec. 3–6.
If the system is nonlinear or time varying, other types of distortion will be produced. As
a result, there are new frequency components at the output that are not present at the input. In
some communication applications, the new frequencies are actually a desired result and, consequently,
might not be called distortion. The reader is referred to Sec. 4-3 for a study of these
effects. In Sec. 4–5, the time delay of bandpass filters is studied, and formulas for group delay
and phase delay are developed.
2–7 BANDLIMITED SIGNALS AND NOISE
A bandlimited waveform has nonzero spectra only within a certain frequency band. In this
case we can apply some powerful theorems—in particular, the sampling theorem—to process
the waveform. As shown in Chapter 3, these ideas are especially applicable to digital communication
problems.
First we examine some properties of bandlimited signals. Then we develop the sampling
and dimensionality theorems.