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