Page 357 - Analog and Digital Filter Design
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354 Analog and Digital Filter Design
have a bandwidth limited to 3.4kHz, signals are sampled at 8 kHz. This means
that analog signal frequencies above 4 kHz must be attenuated to levels below
the input noise floor. To achieve this, an analog filter having a very steep skirt
response above the cutoff frequency is required.
Under-Sampling
Under-sampling is when the sampling frequency is less than twice the highest fre-
quency of the analog signal. Under-sampling introduces alias signals into
the passband of the wanted signals, and these cannot be removed. During the
sampling process, the sampling pulse is multiplied by the analog signal in the time
domain. The resultant frequency domain spectrum at the output of this process
is the sum and difference of the sampling frequency and the analog signal.
Aliasing, due to under-sampling, is easily explained by example. Suppose the
telephone system described previously is not filtered very well and allows
through analog signals that have frequency content above half the sampling
frequency. Consider an unwanted signal with a frequency of 6kHz. With a
sampling frequency of 8 kHz, the output signal will have a frequency spectrum
that includes (8 + 6)kHz and (8 - 6)kHz. In other words, 14 kHz and 2 kHz. The
2kHz signal is the problem, since it is within the 3.4kHz passband and cannot
be removed by subsequent processing.
Under-sampling can sometimes serve a useful purpose. Suppose we have a
speech signal with a bandwidth of lOkHz, but it is amplitude modulated on a
carrier at 1 MHz. This signal could be sampled at about 2.5MHz but would
contain a lot of useless information. We want to know about the speech signal,
not the carrier. If the signal is instead sampled at 1.03 MHz, the mixing process
generates signals centered at 30 kHz and 2.03 MHz. The signal could be deci-
mated by 1/8 to sample at 128.75 kHz before being demodulated.
Care must be taken when under-sampling to avoid aliasing unwanted signals
into the passband. The sampling frequency must be more than twice the
bandwidth of the analog signal. However, the sampling frequency may be lower
than the ceizter+equeizcji of the analog signal. The location of spectral images
is given by:
In this equation fi is the frequency of the image, is the sampling frequency,
andJ;. is the analog signal frequency. The image is repeated across the spectrum,
indicated by the multiplying integer n.
In order to avoid destructive aliasing, the following condition must be met:
