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Unformatted text preview: undamental frequency range).
We know t hat a sinusoid of frequency 0 > 7r a ppears as a sinusoid of a lower
frequency 0 ~ 7r. For a sampled continuous-time sinusoid, this fact means t hat
samples of a sinusoid of frequency w > 7r I T a ppear as samples of a sinusoid of
lower frequency w ~ 7r I T. T he mechanism of how t he samples of continuous-time
sinusoids of two (or more) different frequencies can generate t he s ame discretetime signal is shown in Fig. 8.12. T his phenomenon is known a s a liasing because,
through sampling, two entirely different a nalog sinusoids take on t he s ame "discretetime" identity.
Aliasing causes ambiguity in digital signal processing, which makes it impossible t o d etermine t he t rue frequency of t he s ampled signal. Therefore, aliasing
is highly undesirable a nd should b e avoided. To avoid aliasing, t he frequencies
of t he continuous-time sinusoids t o b e processed should be kept within t he range
wT ~ 7r o r w ~ 7r I T. U nder t his condition, t he question of ambiguity or aliasing
does not arise because any continuous-time sinusoid o f frequency in this range has
a unique waveform when it is sampled. Therefore, if W h is t he highest frequency t o
b e processed, then, t o avoid aliasing,
I f:h is t he highest frequency in Hertz, F h = wh/27r, a nd, according t o Eq. (8.17a),
Fh ~ 2T (8.17b) 1
T<- 2Fh (8.17c) or This result shows t hat d iscrete-time signal processing places t he limit on t he highest
frequency F h t hat can b e processed for a given value of the sampling interval T 560 8 Discrete-time Signals a nd Systems 8.4 Useful Signal Operations 561 8 .4-3 Time Scaling I [k] (a) o 6 1I1
10 15 k _ ___ Following t he a rgument used for continuous-time signals, we c an show t hat t o
t ime scale a signal f [k] by a factor a, we replace k w ith ak. However, because
t he discrete-time argument k c an take only integral values, certain restrictions a nd
changes in t he p rocedure are necessary.
T ime Compression: Decimation or Downsampling Consider a signal fc[k] ~[k]=1 [ k-5]
(O.9)k - 5 ( b) 10 0 12 15 k _ ___ 1 ,[k]=1 [ -k] ( O.9f r
-10 I = f[2k] (8.22) T he signal fe[k] is t he signal f [k] compressed by a factor 2. Observe t hat fe[O] =
frO]' fe[l] = f, fe = f, a nd so on. This fact shows t hat fe[k] is m ade
up of even numbered samples of f[k]. T he o dd numbered samples of f [k] a re
missing (Fig. 8.17b).t This operation loses p art of t he d ata, a nd t hat is why such
time compression is called d ecimation or d ownsampling. I n t he continuous-time
case, time compression merely speeds up t he signal without loss of data. In general,
f[mk] (m integer) consists of only every m th sample of f[k].
T ime Expansion k Consider a signal (c) fe[k]=f[~]
-6 -3 0 6 15 k _ ___ F ig. 8 .16 Time-shifting and time inversion of a signal. 8 .4-1 T ime Shifting
Following t he a rgument used for continuous-time signals, we c an show t hat
t o t ime shift a signal f [k] by m units, we replace k with k - m. Thus, f [k - m]
represents f [k] t ime shifted by m units. I...
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