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Fundamentals of discrete-time signal Discrete-time signals:
processing • Signals: physical quantities that change as a function
of time, space, or some other dependent variable.
Objective of this chapter: • Analysis of signals require mathematical signal
To focus attention on some important issues of models that allow one to choose the appropriate
discrete-time signal processing that are of fundamental mathematical approach for analysis.
importance to signal processing. • Signal characteristics and the classification of signals
based upon either such characteristics or the
associated mathematical models are the subject of
this Section.
Continuous-time, discrete-time and digital signals
• real-valued / complex-valued signal : depends on the
value of the dependent variable
• continuous / discrete : every signal variable may take
on values from either a continuous set of values or a
discrete set of values:
Dependent Independent
variable variable
Continuous- Continuous Continuous
time signal
Digital signal Discrete Discrete
Discrete signal Don't care Discrete
• Discrete signal is our concern in this class.
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Mathematical description of signals • Energy signal vs. Power signal:
• The mathematical analysis of a signal requires the Energy Power
availability of a mathematical description for the Energy Finite, 0 < Ex < ∞ =0
signal itself. signal (def)
Power = ∞ Finite, 0 < P < ∞
signal x
• The description is usually referred to as a signal (def)
model. • Energy of a signal : E = ∞ x(n)2 ≥ 0
x ∑
−∞
• In the book, the term signal is used to refer to either • Power of a signal : P = lim 1 N x(n)2 ≥ 0
the signal itself or its model. x N→∞ 2N+1 ∑
−N
Deterministic signals • A discrete-time signal x(n) is periodic with
• Any signal that can be described by an explicit fundamental period N if x(n+N)=x(n) for all N.
mathematical relationship is called deterministic. Otherwise it is called aperiodic.
• Some basic signals: • A periodic signal is a power signal.
• Unit impulse sequence: δ(n)=1 if n=0; δ(n)=0 else • A signal x(n) has finite duration if x(n)=0 for nN2, where N1 and N2 are finite integer
• Exponential sequence of the form: x(n)=an numbers and N1≤N2. If N1=-∞ and/or N2=∞, it has
infinite duration.
• Signal classification: causal
Deterministic signals can be classified as (1) energy • A signal x(n) is said to be if x(n)=0 for n<0.
or power, (2) periodic or periodic, (3) of finite or Otherwise it is noncausal.
infinite duration, (4) causal or non-causal, and (5) even
even or odd signals. • A real-valued signal x(n) is if x(-n)=x(n) and
odd if x(-n)=-x(n).
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Transform-domain representation of deterministic
Random signals signals
• Signals that can't be described to any reasonable • Deterministic signals are assumed to be explicity
accuracy by explicit mathematical relationships are known for all time, so the simplest description of any
called random signals. signal is an amplitude-versus-time plot.
• Though random signals are evolving in time in an • Frequency analysis is the process of decomposing a
unpredictable manner, their average properties can signal into frequency components.
often be assumed to be deterministic.
• Random signals are thus mathematically described by • Two characteristics that specifies the analysis tools:
stochastic processes and can be analyzed by using (1) The nature of time: continuous-time or discrete-
statistical methods instead of explicit equations. time signals.
(2) The existence of harmony: periodic or aperiodic
• The theory of probability, random variables, and signals
stochastic processes provides the mathematical
framework for the theoretical study of random Periodic Aperiodic
signals. Continuous- Fourier series Fourier Transform
time 1T ∞ − π
− π j ft
X k = x t e j2 kt/Tdt X f = x t e 2 dt
Real-world signals: ( ) T ∫ ( ) ( ) ∫ ( )
0 t=−∞
∞ π ∞
j2 kt/T j ft
x t = X k e 2π
• In practical terms, the decision as to whether physical ( ) ∑ ( ) x(t) = ∫ X( f )e df
data are deterministic or random is usually based k=−∞ f =−∞
upon the ability to reproduce the data by controlled Discrete- Fourier series (c.w. DFT) Fourier transform
N− ∞
time 1 1 j N kn ω − ω
X = ∑ x(n)e− (2π/ ) X(ej ) = ∑x(n)e j n
experiments. k N n=0 n=−∞
N− π
1 j(2π/ N)kn 1 ω ω
x(n) = X e x n X ej ej nd
∑ k ( ) = 2π ∫ ( ) ω
k=0 −π
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• Spectral classification:
Sampling of continuous-time signals
• In most practical applications, discrete-time signals
are obtained by sampling continuous-time signals • Sampling theorem:
periodically in time. In order to avoid aliasing, the sampling rate
• Sampling frequency/rate: the number of samples must be at least equal to twice the bandwidth of a
taken per unit of time (=F ) band-limited, real-valued, continuous-time signal.
s
• The minimum sampling rate of F = 2B is called
• Sampling period: 1/F s
s Nyquist rate.
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The discrete Fourier transfrom • The contour of integration in the inverse z-transform
• The discrete Fourier transform (DFT) of a sequence can be any counterclockwise closed path that
x(n) and the corresponding inverse discrete Fourier encloses the origin and is inside the ROC.
transform (IDFT) are, respectively, given by: • Connection between z-transform & DFT:
ω
X(z)| = X(ej )
N−1 − j(2π/N)kn x=e jω
Xk = ∑ x(n)e
n=0
x(n) = 1 N−1X ej(2π/N)kn • Properties of z-transform
N ∑ k
k=0
(c.w. the analysis tool for discrete-time periodic
deterministic signal.)
The z-transform
• The z-transform of a sequence x(n) and the
corresponding inverse z-transform are, respectively,
given by:
X z ≡ Ζ x n = ∞ x n z−n (2.2.29)
( ) [ ( )] ∑ ( )
n=−∞
1 n−
x(n) = ∫ X(z)z 1dz
2 j
π C
• The set of values of z for which (2.2.29) converges is
called the region of convergence (ROC) of X(z).
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Discrete-time systems Time-domain analysis
• In this section, we review the basics of linear, time- • The output of a linear, time invariant system can
invariant systems. always be expressed as the convolution summation
• A system is defined to be any physical device or between the input sequence x(n) and the impulse
response sequence h(n) of the system.
algorithm that transform a signal, called the input or y(n) = h(n)*x(n) = ∞ x(k)h(n−k)
excitation, into another signal, called the output or ∑
response. k=−∞
• The mathematical relationships between the input and • In matrix form, we've
output signals of a system is referred to as a system x(0) 0 0
y(0)
model.
0 h(0)
y(M −1) x(M −1) x(0)
x(n) H(z) y(n) h(1)
=
Block diagram representation of a discrete-time system y(N −1) x(N −1) x(N−M)h(M −1)
0
y(L−1)
0 0 x(N−1)
Analysis of linear, time-invariant (LTI) systems
y(0) h(0) 0 0 x(0)
• The systems we deal with are linear and time- h(1) h(0) 0
invariant and are always assumed to be initially at or y(1) = x(1)
rest. h M −
y(L−1) 0 0 ( 2)x(N −1)
0 0 h(M−1)
Toeplitz matrix: all the elements along any
diagonal are equal.
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• A system is called causal if the present value of the Transform domain analysis
output signal depends only on the present and/or past • y(n) = h(n)* x(n) ⇔ Y(z)=H(z)X(z)
values of the input signal. (i.e. h(n)=0 for n<0)
• A system is called stable if every bounded input • A causal discrete-time system can also be described
produces a bounded output. with a linear difference equation.
Q
(i.e. x(n) < ∞ ⇒ y(n) < ∞ for all n) P
y(n)ay(nk)dx(nk)
=−−+−
∑∑
kk
==
kk
∞ 10
• An LTI system is stable iff <∞.
∑|h(n)|
n=−∞ • If system parameters {ak,dk} depend on time, the
• A system has an impulse response with finite duration system is time-varying. Otherwise, it's time-invariant.
is called a finite impulse response (FIR) system. • If system parameters {a ,d }depend on either the
Otherwise, it's called an infinite impulse response k k
(IIR) system. input or output signals, the system is nonlinear.
Otherwise, it's linear.
• With z-transform, we've
Q d z−k
Y(z) ∑ k D(z)
H(z) = = k=0 ≡
X(z) 1+ P a z−k A(z)
∑ k
=
k 1
It can be rewritten as
Q −
∏(1−z z 1)
D(z) k= k
H(z)= =G 1
A z P
( ) −
∏(1−p z 1)
k= k
1
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• The roots of D(z) and A(z) are, respectively, referred All-pole (AP) system: (Q=0)
to as zeros and poles of the system.
• y(n) = − P a y(n−k)+ x(n)
∑ k
k=1
1 P A
• H(z) = = ∑ k
P −k k=1 p z−k
1+ ∑a z 1− k
k= k
1
• h(n) = P A (p )nu(n)
∑ k k
k=1
• Any nontrivial pole in a system implies an infinite
• The system is stable if its poles are all inside the unit duration impulse response.
cycle.
All-zero (AZ) system: (P=0)
• y(n) = Q d x(n−k)
∑ k
k=0
• H(z) = Q d z−k
∑ k
k=0
• h(n) = dn 0 ≤ n ≤ Q
0 elsewhere
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