7.4 Practical Aspects of Sampling 439
nExample 7.4
Consider the development of an AM transmitter that uses a computer to generate the modulated
signal and is capable of transmitting music and speech signals. Indicate how to implement the
transmitter.
Solution
Let the message bem(t)=x(t)+y(t)wherex(t)is a speech signal andy(t)is a music signal. Since
music signals display larger frequencies than speech signals, the maximum frequency ofm(t)is
that of the music signals, orfmax=22 KHz. To transmitm(t)using AM, we modulate it with a
sinusoid of frequencyfc>fmax, sayfc= 3 fmax=66 KHz.
To satisfy the Nyquist sampling rate condition, the maximum frequency of the modulated sig-
nal would befc+fmax=( 66 + 22 )KHz=88 KHz, and so we would chooseTs= 10 −^3 / 176
sec/sample as the sampling period. However, according to the above results we can also choose
Ts= 1 /( 2 B)whereBis the bandwidth ofm(t)in hertz orB=fmax=22 KHz, which gives
Ts= 10 −^3 /44 — four times larger than the previous sampling period, so we choose this as the
sampling period.
The analog signalm(t)to be transmitted is inputted into an ADC in the computer, capable of
sampling at 44, 000 samples/sec. The output of the converter is then multiplied by a computer-
generated sinusoid
cos( 2 πfcnTs)=cos( 2 π× 66 × 103 ×( 10 −^3 / 44 )n)=cos( 3 πn)=(− 1 )n
to obtain the AM signal. The AM digital signal can then be inputted into a DAC and its output sent
to an antenna for broadcasting. n
7.4 Practical Aspects of Sampling
To process analog signals with computers it is necessary to convert analog into digital signals and
digital into analog signals. The analog-to-digital and digital-to-analog conversions are done by ADCs
and DACs. In practice, these converters differ from the ideal versions we have discussed so far where
the sampling is done with impulses, the discrete-time samples are assumed representable with infi-
nite precision, and the reconstruction is performed by an ideal low-pass filter. Pulses rather than
impulses are needed, and the discrete-time signals need to be discretized also in amplitude and the
reconstruction filter needs to be reconsidered.
7.4.1 Sample-and-Hold Sampling
In an actual ADC the time required to do the sampling, quantization, and coding needs to be con-
sidered. Therefore, the width 1 of the sampling pulses cannot be zero as assumed. Asample-and-hold
sampling systemtakes the sample and holds it long enough for quantization and coding to be done
before the next sample is acquired. The question is then how does this affect the sampling process
and how does it differ from the ideal results obtained before? We hinted at the effects when we
considered the PAM before, except that now the resulting pulses are flat.
