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frequency content in the frequency domain in integer multiples of sampling frequency, it
results in adding two very close frequency contents. The result is that the sampled signal
shows the beat phenomenon (Figure 2.14). Another example of the same phenomenon is
y(t) =sin(2 4.9t) (2.35)

and sample it with a sampling frequency of w = 10.0Hz.
s
In Figure 2.14, the top-left figure shows the frequency content of the original sig-
nal, and the top-right figure shows the frequency content of the sampled signal. If we
approximate the sampled signal with the lowest two frequency contents,
1
∗
y (t) ≈ (sin(2 0.9t) +sin(2 1.0t)) (2.36)
T
which is the addition of two sinusoidal signals with very close frequencies, as discussed
above. Therefore, the sampling is expected to have (and the above equation explains) the
beat phenomenon observed in the time domain plots of the sampled signal.

2.2.4 Signal Reconstruction: D/A Operation

A D/A converter is used to convert a digital number to an analog voltage signal. It is also
referred to as the signal reconstruction device. Let us consider that we sample a continuous
signal through an A/D converter, then send that signal out without any modification through
a D/A converter. The difference between the original analog signal (input to the A/D
converter) and the analog signal output from the D/A converter is an undesired distortion due
to sampling, quantization, time delay, and reconstruction errors (Figure 2.1). For instance,
in communication systems, the analog voice signal is sampled, transmitted digitally over
the phone lines, and converted back to the analog voice signal at the other end of the phone
line. The goal there is to be able to reconstruct the original signal as accurately as possible.
We know that the sampled signal frequency content is the original frequency content
plus the same content shifted in the frequency axis by integer multiples of the sampling
frequency. In order to recover the frequency content of the original signal, we need an ideal
filter which has a square gain and zero phase transfer function (Figure 2.7). Clearly, if there

DAC
y(t) y(kT) Ideal
ADC
Reconstruction
Filter
T(s)
ω = π
2 /T (rad/s)
S
t 0T 2T..... t
F F
Y 0
|y( j )|ω
/T
y 0 ω ω
- S -ω ω S
-ω -ω ω ω ω |H( j )|ω 2 1 1 2
S 1 1 S
T Y 0
-ω ω ω
1 1
ω ω ω ω
- 2 S 2 S - 2 S 2 S
ω
FIGURE 2.7: Signal reconstruction with an ideal D/A converter.