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Discrete-Time Signals
Week 2
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Continuous-Time Versus Discrete-Time Signals
Continuous-time signal: a signal defined by a function of a continuous-time variable.
Discrete-time Signal: a signal defined by specifying the value of the signal only at discrete times, called sampling instants.
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Continuous-Time Versus Discrete-Time Signals
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Distinction between Discrete-Time and Quantized Signals
Quantized Signal: A quantized signal is one whose values may assume only a countable number of values, or levels, but the changes from level to level may occur at any time.
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Periodic and Aperiodic Signals
Periodic: a signal x(t) is periodic if and only if
x(t+T0)=x(t), -<t< (1)
where the constant T0 is the period. Aperiodic: Any deterministic signal
not satisfying (1) is called aperiodic.
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Periodic Signals
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Aperiodic Signals
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Some Elementary Discrete-Time Signals
Unit sample:
Unit step:
otherwise
nn
0
01
otherwise
nnu
0
01
n
k
knu
1 nunun
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Some Elementary Discrete-Time Signals
Exponential sequence
Sinusoidal sequence
nAnx
nAnx cos
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Some Elementary Discrete-Time Signals
(1) Unit sample (2) Unit step
(3) Exponential sequence
(4) Sinusoidal sequence
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Energy and Power Signals
Total energy Average power
1. X(t) is an energy signal if and only if 0<E<, so that P=0.
2. X(t) is an power signal if and only if 0<P<, thus implying that E=.
3. Signals that satisfy neither property are therefore neither energy nor power signals.
joulesdttxET
TT
2
lim
wattsdttxT
PT
TT
2
2
1lim
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Addition, Multiplication, and Scaling
Addition
Multiplication
Scaling
,...2,1,0)()()( 21 nnxnxny
,...2,1,0)()()( 21 nnxnxny
,...2,1,0)()( nncxny
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Convolution of Two Discrete-Time Signals
Convolution Properties
1. Commutative property
2. Associative property
3. Distributive property
k
knhkxnhnx
nxnhnhnx
nhnhnxnhnhnx 2121
nhnxnhnxnhnhnx 2121
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Correlation of Two Discrete-Time Signals
Cross-correlation function
Autocorrelation function
p=0 maximum
1
0
1lim
N
nNxy pnynx
Npc
1
0
1lim
N
nNxx pnxnx
Npc