1. Introduction
2. Block Diagram representation
3. Convolution Sum
4. LTI system analysis using DTFT
5. LTI system analysis using Z transform

**LINEAR TIME INVARIANT DISCRETE
TIME SYSTEMS**

**1. Introduction**

A
discrete-time system is anything that takes a discrete-time signal as input and
generates a discrete-time signal as output.1 The concept of a system is very
general. It may be used to model the response of an audio equalizer . In
electrical engineering, continuous-time signals are usually processed by
electrical circuits described by differential equations.

For
example, any circuit of resistors, capacitors and inductors can be analyzed
using mesh analysis to yield a system of differential equations. The voltages
and currents in the circuit may then be computed by solving the equations. The
processing of discrete-time signals is performed by discrete-time systems.
Similar to the continuous-time case, we may represent a discrete-time system
either by a set of difference equations or by a block diagram of its
implementation.

For
example, consider the following difference equation. y(n) =
y(n-1)+x(n)+x(n-1)+x(n-2) This equation represents a discrete-time system. It
operates on the input signal x(n)x(n) to produce the output signal y(n).

**2. BLOCK DIAGRAM REPRESENTATION**

Block
diagram representation of

LTI
systems with rational system function can be represented as
constant-coefficient difference equation

• The
implementation of difference equations requires delayed values of the

– input

– output

–
intermediate results

• The
requirement of delayed elements implies need for storage

We also
need means of

–
addition

–
multiplication

**Direct Form I**

General
form of difference equation

Alternative
equivalent form

• Cascade
form

General
form for cascade implementation

Parallel
form

Represent system function using partial
fraction expansion

**3. CONVOLUTIO N SUM**

The
convolution sum provides a concise, mathematical way to express the output of an
LTI system based on an arbitrary discrete-time input signal and the system's
response. The convolution sum is expressed as

LTI
systems are causal if

h[n]
= 0 n < 0

**4. LTI System
analysis using DTFT**

**LTI SYSTEMS ANALYSIS USING DTFT**

**5. LTI SYSTEM
ANALYSIS USING Z-TRANSFORM**

The
z-transform of impulse response is called transfer or system function H(z).

Y(z)=X(z)H(z)

General
form of LCCDE

System
unction: Pole/zero Factorization

Stability
requirement can be verified

Choice
of ROC determines causality.

Location
of zero and poles determines the frequency response and phase

**Sample Problems:**

**1. Consider the system described by
the difference equation.**

**2. Given y[-1]=1 and y[-2]=0.
Compute recursively a few terms of the following 2 ^{nd} order DE:**

**3. Compute the impulse response of
the system described by,**

**4. Obtain the structures
realization of LTI system**

**5. Find the convolution of
x(n)=[1,1,1,1,2,2,2,2] with h(n)=[3,3,0,0,0,0,3,3] by using matrix method.**

**Solution: By using matrix method,
N=8**

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Signals and Systems : Linear Time Invariant Discrete Time Systems : Linear Time Invariant Discrete Time Systems |

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