Discrete-Time Signals Week 2
Dec 21, 2015
Discrete-Time Signals
Week 2
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.
Continuous-Time Versus Discrete-Time Signals
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.
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.
Periodic Signals
Aperiodic Signals
Some Elementary Discrete-Time Signals
Unit sample:
Unit step:
otherwise
nn
0
01
otherwise
nnu
0
01
n
k
knu
1 nunun
Some Elementary Discrete-Time Signals
Exponential sequence
Sinusoidal sequence
nAnx
nAnx cos
Some Elementary Discrete-Time Signals
(1) Unit sample (2) Unit step
(3) Exponential sequence
(4) Sinusoidal sequence
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
Addition, Multiplication, and Scaling
Addition
Multiplication
Scaling
,...2,1,0)()()( 21 nnxnxny
,...2,1,0)()()( 21 nnxnxny
,...2,1,0)()( nncxny
Convolution of Two Discrete-Time Signals
Convolution Properties
1. Commutative property
2. Associative property
3. Distributive property
k
knhkxnhnx
nxnhnhnx
nhnhnxnhnhnx 2121
nhnxnhnxnhnhnx 2121
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