Discrete-time Signals

ELEC 423Prof. Siripong Potisuk

Mathematical Representation

x[n] represents a DT signal, i.e., a sequence of numbers defined only at integer values of n

(undefined for noninteger values of n) Each number x[n] is called a sample x[n] may be a sample from an analog signal xd[n] = xa(nTs), where Ts = sampling period

Discrete-time Signals

Discrete-time, continuous-valued amplitude (sampled-data signal)

Discrete-time, discrete-valued amplitude (digital signal)

In practice, we work with digital signals

Signal Transformations

Operations performed on the independent and the dependent variables

1) Reflection or Time Reversal or Folding2) Time Shifting3) Time Scaling4) Amplitude Scaling5) Amplitude ShiftingNote: The independent variable is assumed to be n representing sample number.

Time reversal or Folding or Reflection

][][ nxny

For DT signals, replace n by –n, resulting in

y[n] is the reflected version of x[n] aboutn = 0 (i.e., the vertical axis).

Time Shifting (Advance or Delay)

(Advance) by left the toshifted is ][ I

(Delay) by right the toshifted is ][ I

],[][ in resultingshift ofamount theis here w

by replace :Signals DTFor

00

00

0

0

0

nnxn

nnxn

nnxnyn

nnn

Time Scaling

Sampling rate alteration

][ of that )(1/ is ][ of rate sampling the

],[][ : sampling-Downth nxkny

Ikknxny

][ of that times is ][ of rate sampling the otherwise ,0integeran is n/D if ],/[

][ : sampling-Up

nxDny

Dnxny

Amplitude Scaling

][][ nAxny

For DT signals, multiply x[n] by the scalingfactor A, where A is a real constant.

If A is negative, y[n] is the amplitude-scaledand reflected version of x[n] about thehorizontal axis

Amplitude Shifting

Anxny ][][

For DT signals, add a shifting factor A to x[n], where A is a real constant.

Signal Characteristics

Deterministic vs. Random

Finite-length vs. Infinite-length

Right-sided/ Left-sided/ Two-sided

Causal vs. Anti-causalPeriodic vs. Aperiodic (Non-periodic)Real vs. ComplexConjugate-symmetric vs. Conjugate-antisymmetricEven vs. Odd

Even & Odd DT Signals

].[][ if ricantisymmet Conjugate

],[][ if symmetric Conjugate*

*

nxnx

nxnx

A real-valued signal x[n]) is said to be

A complex-valued sequence x[n] is said to be

].[][ if odd],[][ ifeven nxnxnxnx

DT Periodic Signals

A DT signal x[n] is said to be periodic if

IknkNnxNnxnx , ],[][][ 0

N0 is the smallest positive integer called the fundamental period (dimensionless)

N0 = the fundamental angular frequency (radians)

DT Sinusoidal Signals

]cos[][ 0 nAnx

n is dimensionless (sample number)

and have units of radians

x[n] may or may not be periodicperiodic if 0 is a rational multiple of 2,i.e.,

period lfundamenta the and ,2 00

0 NImNm

Discrete-time periodic sinusoidal sequences for several different frequencies

An example of a discrete-time non-periodic sinusoidal sequence

DT Unit Impulse

DT Unit impulse (Kronecker delta function)

Properties:

0,10,0

][nn

n

][][][)4

][][][][ )3

][][)2 1][)1

00

000

n

n

nxnnnx

nnnxnnnx

nnn

Unit Step Function

0,10,0

][nn

nu

u [n] can be written as a sum of shifted unitimpulses as

0

][][k

knnu

Exponential Signals

][

, lettingBy

numberscomplex or real are and where][

n

n

Benx

er

rBBrnx

Signal Metrics

EnergyPowerMagnitudeArea

Energy

2][n

x nxE

An infinite-length sequence with finite sample values may or may not have finite energy

Power

N

NnNx nx

NP 2|][|

121lim

- A finite energy signal with zero average power is called an ENERGY signal- An infinite energy signal with finite average power is called a POWER signal

Average Power

Average over one period for periodic signal, e.g.,

Root-mean-square power:

0

12 any for |][|1 0

0

nnxN

PNn

nnx

xrms PP

Magnitude & Area

x

nx

M

nxM

if Bounded

][max :Magnitude

n

x nxA ][ :Area