Discrete-time Signals
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Discrete-time Signals ELEC 423 Prof. Siripong Potisuk
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Discrete-time Signals. ELEC 423 Prof. 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 - PowerPoint PPT Presentation
Transcript of Discrete-time Signals
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
