EEE 503
Digital Signal Processing
Lecture #2 : Discrete-Time Signals & Systems
Dr. Panuthat Boonpramuk
Department of
Control System & Instrumentation Engineering
KMUTT
Analog Signal & Digital Signal
Analog Signal
x(t)
Sampling
Discrete Time Signal
x(nT), x(n)
Quantization
Digital Signal
X(n)
Analog Signal & Digital Signal
Discrete-Time Signals : Sequences
Discrete-time (Digital) signals are represented mathematically as
sequences of numbers.
x={x[n]}, −∞ < n < ∞
where n is an integer.
In practice, such sequences can often arise from periodic sampling of an
analog signal.
x[n] =xa[nT], −∞ < n < ∞
where T is called the sampling period, and fs=1/T is the sampling rate.
Basic Sequences (1)
Basic Sequences (2)
Basic Sequences (3)
Basic Sequences (4)
Periodicity of Sinusoidal Sequences and
Complex Exponential Sequence
Sinusoidal Sequences with Different Frequencies
Discrete-Time Systems (1)
 A discrete-time system is defined mathematically as a transformation or
operator that maps an input sequence x[n] into a unique output sequence
y[n].
Discrete-Time Systems (2)
 The ideal delay system
 Memoryless systems
A system is referred as memoryless system if the output y[n] at every value
of n only depends on the input x[n] at the same value of n. For example,
Discrete-Time Systems (3)
 Linear systems
The class of linear systems is defined by the principle of superposition. If
and a is an arbitrary constant, then the system is linear if and only if
(additivity property)
(homogeneity or scaling property)
Discrete-Time Systems (4)
 Time-invariant systems
A time-invariant system is a system for which a time shift or delay of the
input sequence causes a corresponding in the output sequence.
Discrete-Time Systems (5)
 Causality
A system is causal if, for every choice of n0, the output sequence value at
the index n=n0 depends only on the input sequence value for n ≤ n0.
 Examples
Forward difference system – not causal
y[n] = x[n+1] - x[n]
Backward difference system - causal
y[n] = x[n] - x[n-1]
Discrete-Time Systems (6)
 Stability
A system is in the bounded input, bounded output (BIBO) sense if and
only if every bounded input sequence produces a bounded output
sequence. The input x[n] is bounded if there exists a fixed positive finite
value Bx such that
|x[n]| ≤ Bx <∞ for all n.
Stability requires that, for every bounded input, there exists a fixed
positive finite value By such that
|y[n]| ≤ By <∞ for all n.
Linear Time-Invariant Systems
 A particularly important class of systems consists of those that are linear
and time-invariant (LTI).
Impulse
If
Time-invariant
Linear
response
Convolution Sum (1)
 Convolution Sum
Example
Convolution Sum (2)
 Method 1
1. For each k for which x[k] has a nonzero value, evaluate x[k] h[n–k]
corresponding to the specific x[k]. It equals to the waveform of h[n]
multiplied by x[k] and timeshifted by k (shift toward right if k>0, and shift
toward left if k<0).
2. Add the resultant sequence values for all k’s to obtain the convolution
sum corresponding to the full input sequence x[n].
Convolution Sum (3)
 Method 1
Convolution Sum (4)
 Method 2
1. For each value n (see *), producing h[n – k]. This is the mirror image of h[k]
about the vertical axis shifted by n (shift toward right if n>0, and shift toward
left if n<0).
2. Multiply this shifted sequence h[n–k] and the input sequence x[k], and add
the resultant sequence values to obtain the value of the convolution at n.
3. Repeat steps 1-2 for different value of n.
[* Note the range of n: if x[n] has its nonzero value between x1 and x2, and h[n]
has nonzero values between h1 and h2, then x[n]*h[n] has nonzero value
between x1+h1 and x2 +h2.]
Convolution Sum (5)
 Method 2
Properties of LTI Systems (1)
The impulse response is a complete characterization of the properties of a
specific LTI system.
 Convolution operation is commutative
x[n]*h[n] = h[n]*x[n]
 Parallel combination of LTI systems
x[n]*(h1[n]+h2[n]) = x[n]*h1[n]+x[n]*h2[n])
Properties of LTI Systems (2)
 Cascade connection of LTI systems
h[n]= h1[n]*h2[n]
Properties of LTI Systems (3)
 Stability
LTI systems are stable if and only if the impulse response is absolutely
summable, i.e, if
 Causality
LTI systems are causal if and only if
Impulse Responses of Some LTI Systems
 Ideal delay (stable, causal when nd ≥ 0)
 Accumulator (unstable, causal)
 Forward difference system (stable, noncausal)
 Backward difference system (stable, causal)
Inverse System
If a LTI system has impulse response h[n], then its inverse system,
if exists, has impulse response hi[n] defined by the relation
 Example
Linear Constant-Coefficient Difference Equations (1)
 An important subclass of LTI systems consists of those systems
for which the input x[n] and the output y[n] satisfy an Nth-order
linear constant-coefficient difference equation of the form
present & past
inputs
If a0=1, then
presen
t
output
past
outputs
present & past
inputs
Linear Constant-Coefficient Difference Equations (2)
 Recursive filter
At least one ak≠0 (k = 1, …, N). h[n] has infinite support. Also
known as infinite impulse response (IIR) filter.
 Non-recursive filter
a1, …, aN =0 (no feedback). h[n] has finite support. Also known
as finite impulse response (FIR) filter.
 Example - accumulator
Recursive Computation of Difference Equations (1)
Recursive Computation of Difference Equations (2)
Recursive Computation of Difference Equations (3)
 For a system defined by an Nth-order linear constant-coefficient
difference equation, the output for a given input is not uniquely
specified. Auxiliary information or conditions are required.
 If the auxiliary information is in the form of N sequential values
of the output, then the output of the system is uniquely specified.
 Linearity, time-invariance, and causality of the system depend
on the auxiliary conditions. If an additional condition is that the
system is initially at rest, then the system will be LTI and causal.
Frequency-Domain Representation of
Discrete-Time Signals and Systems (1)
Frequency-Domain Representation of
Discrete-Time Signals and Systems (2)
Eigenfunction and eigenvalue
A signal for which the system output is just a (possibly complex)
constant times the input is referred to as an eigenfunction of the
system, and the constant factor is referred to as the eigenvalue.
 Consider the cases that the input signals are complex
exponential sequences. Complex exponential sequences are
eigenfunctions of LTI systems. The response to a complex
exponential sequence input is complex exponential sequence with
the same frequency as the input and with amplitude and phase
determined by the system.
Frequency-Domain Representation of
Discrete-Time Signals and Systems (3)
Frequency-Domain Representation of
Discrete-Time Signals and Systems (4)
Frequency-Domain Representation of
Discrete-Time Signals and Systems (5)
Frequency-Domain Representation of
Discrete-Time Signals and Systems (6)
Fourier Representation
 Fourier transform
 Inverse Fourier transform
 X(ejω) is in general a complex function of ω.
Fourier Representation – Examples (1)
 Fourier transform (real and imaginary parts)
Fourier Representation – Examples (2)
 Fourier transform (magnitude and phase)
Fourier Representation – Examples (3)
 Fourier transform using normalized frequency (fs=22 kHz)
Fourier Representation – Examples (4)
 Fourier transform using actual frequency (fs=22 kHz)
Fourier Representation – Examples (5)
 Fourier transform using normalized frequency
Fourier Representation – Examples (6)
 Fourier transform using actual frequency