EEE 503EEE 503 Digital Signal Processing Digital Signal Processing

Lecture #2 : Lecture #2 : Discrete-Time Signals & Systems

Dr. Panuthat BoonpramukDepartment of

Control System & Instrumentation Engineering

KMUTT

Analog Signal & Digital Signal

Analog Signalx(t)

Discrete Time Signalx(nT), x(n)

Digital SignalX(n)

Sampling

Quantization

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].

The ideal delay system

Memoryless systemsA 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 (2)

Linear systems The class of linear systems is defined by the

principle of superposition. If

Discrete-Time Systems (3)

and a is an arbitrary constant, then the system is linear if and only if

(additivity property)

(homogeneity or scaling property)

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 (4)

CausalityA 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.

Discrete-Time Systems (5)

ExamplesForward 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).

If

Time-invariant

Linear

Impulse response

Convolution Sum (1) Convolution Sum

Example

Convolution Sum (2)

Method 11. 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 1Method 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 2Method 2

Properties of LTI Systems (1)

The impulse response is a complete characterization of the properties of a specific LTI system.

Convolution operation is commutativex[n]*h[n] = h[n]*x[n]

Parallel combination of LTI systemsx[n]*(h1[n]+h2[n]) = x[n]*h1[n]

+x[n]*h2[n])

Properties of LTI Systems (2)

Cascade connection of LTI systemsh[n]= h1[n]*h2[n]

Properties of LTI Systems (3)

StabilityLTI systems are stable if and only if the impulse response is absolutely summable, i.e, if

CausalityLTI 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 SystemIf 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

If a0=1, then present & past inputs

present output

pastoutputs

present & past inputs

Linear Constant-Coefficient Difference Equations (2)

Recursive filterAt least one ak≠0 (k = 1, …, N). h[n] has infinite support. Also known as infinite impulse response (IIR) filter.

Non-recursive filtera1, …, 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