UNIT V Linear Time Invariant Discrete-Time Systems

DT Unit-Impulse Response Consider the DT SISO system: If the input signal is and the system has no energy at , the output is called the impulse response of the system System System

Example Consider the DT system described by Its impulse response can be found to be

Representing Signals in Terms of Shifted and Scaled Impulses Let x[n] be an arbitrary input signal to a DT LTI system Suppose that for This signal can be represented as

Exploiting Time-Invariance and Linearity

The Convolution Sum This particular summation is called the convolution sum Equation is called the convolution representation of the system Remark: a DT LTI system is completely described by its impulse response h[n]

Block Diagram Representation of DT LTI Systems Since the impulse response h[n] provides the complete description of a DT LTI system, we write

The Convolution Sum for Noncausal Signals Suppose that we have two signals x[n] and v[n] that are not zero for negative times (noncausal signals) Then, their convolution is expressed by the two-sided series

Example: Convolution of Two Rectangular Pulses Suppose that both x[n] and v[n] are equal to the rectangular pulse p[n] (causal signal) depicted below

The Folded Pulse The signal is equal to the pulse p[i] folded about the vertical axis

Sliding over

Sliding over - Cont’d

Plot of

Properties of the Convolution Sum Associativity Commutativity Distributivity w.r.t. addition

Properties of the Convolution Sum - Cont’d Shift property: define Convolution with the unit impulse Convolution with the shifted unit impulse then

Direct Form II For a discrete-time system described by the transfer function is of the form

Direct Form II where

Direct Form II

Direct Form II

Direct Form II For the special case of FIR filters. (The number of delays has been changed to M - 1 to conform to conventions in the DSP literature.) This type of filter has M - 1 finite zeros and M - 1 poles at z = 0.

Direct Form II One desirable characteristic of an FIR filter is that it can have linear phase in its passband.

Direct Form II

Direct Form II

Direct Form II

Direct Form II

Direct Form II It can be shown that symmetric or anti-symmetric, even or odd impulse responses yield linear phase shift in the frequency response.

Cascade Realization Direct Form II is no the only form of realization. There are several other forms. Two other important forms are the cascade form and the parallel form. Each individual factor is realized as a small Direct Form II subsystem and the subsystems are then cascaded.

Parallel Realization Each individual term is realized as a small Direct Form II subsystem and the subsystems are then paralleled.

Complex Poles and Zeros In either the cascade or parallel realization, the first-order subsystems may have complex poles and/or zeros. In such a case two first-order subsystems should be combined into one second- order subsystem to avoid the problem of complex coefficients in the first-order subsystems. Also, for reasons we will soon see, it is common to do cascade and parallel realizations with second-order subsystems even when the poles and/or zeros are real.

1st Order vs 2nd Order In the case of FIR filters the second-order subsystems take this form. 2 delays 3 multiplications 2 additions Compare this two two first-order cascaded stages. 2 delays 4 multiplications 2 additions

1st Order vs 2nd Order If the FIR filter has linear phase, a fourth-order structure reduces number of multiplications even further compared with cascaded first-order or cascaded second-order subsystems.