Analysis of Linear Time Invariable Systems

Linear Time Invariant (LTI) Systems

• Linearity – Linear system is a system that possesses the property of superposition.

• Time Invariance – A system is time invariant if the behavior and characteristics of the system are fixed over time.

Discrete – Time LTI systems : The Convolution Sum

• Representation of Discrete – Time Signals in term of Impulses :

Where [n-k] represents unit impulse at n=k

• The Convolution Sum Representation :

[n-k] hk[n]

x[n] y[n]

y[n] = T = =

System T{.}

System T{.}

∑𝑘=−∞

∞

𝑥 [𝑘 ]δ [𝑛−𝑘 ]∑𝑘=−∞

∞

𝑥 [𝑘 ]T {δ [𝑛−𝑘 ]}∑𝑘=−∞

∞

𝑥 [𝑘 ] hk [ n ]¿¿

• Since the system is Time Invariant , then responses to time-shifted unit impulses are all time-shifted versions of each other.

i.e. hk[n] = h0[n-k] and this unit impulse response can be written simply as h[n] for input impulse

• Therefore, for an LTI System

i.e. the convolution or superposition sum which can also be represented as

y[n] = x[n] y[n]

Continuous– Time LTI systems : The Convolution Integral

• A similar approach can be drawn for continuous time LTI systems and following results can be derived.

y(t) = ∫ x(T)h(t-T)dTOr,

y(t) = x(t) h(t)

∞

−∞

Properties of LTI systems :

Commutative x[n] h[n] = h[n] x[n]Distributive x[n] (h1[n]+h2[n]) = x[n] h1[n] + x[n] h2[n]Associative x[n] (h1[n] h2[n]) = (x[n] h1[n]) h2[n]

Stability for LTI Systems :

• An LTI system is stable if every bounded input produces a bounded output.

Consider, |x[n]| < B for all n

Therefore,

Therefore,

i.e.

The output should be bounded for stability

Causality for LTI Systems

• A system is casual if the output at any time depends only on the values of the input at the present time and in the past.

• Therefore, for LTI systems, y[n] must not depend upon x[k] for k > n.

• Hence, h[n] = 0 for n < 0

Z - Transform

• Introduction• Definition• Region of Convergence and Z Plane• Pole and Zero• Example• Properties

Introduction

• Since Fourier Transform has its limitations, a counterpart of Laplace transform (Continuous time) was needed for Discrete time systems.

• To perform transform analysis of unstable systems and to develop additional insight and tools for LTI systems anlysis.

Mathematical Representation

• For a Discrete time LTI system with Impulse Response = h[n] Input = zn

Output Response = y[n] = H(z)zn

Where

Mathematical Representation

• The Z-transform of a general discrete-time signal x[n] is defined as

And represented as, x[n] z X(z)

Z Transform and Discrete time Fourier Transform

• Replace Z = rejω

where r = magnitude ω = angle of ZThe z- transform reduces to the Fourier transform when the magnitude of the transform variable z is unity.• The basic idea is to represent and analyze the

whole system about a unit circle in Z Plane.

Z = ejω

ω

Region of Convergence

• Z transform of a sequence has associated with it a range of values of z for which X(z) converges. This range of values is referred to as the region of convergence.

• A stable system requires the ROC of z-transform to include the unit circle.

Pole and Zero• When X(z) is an rational function, then

1.The roots of the numerator polynomial are referred to as the zeros of X(z).

2.The roots of the denominator polynomial are referred to as the poles of X(z).

• No poles of X(z) can occur within the region of convergence since the z-transform does not converge at a pole.

• The region of convergence is bounded by poles.

Example

• X[n] = anu[n]

ROC is the range of values for which |az-1|< 1Or, |z| > |a|

X(z) = = , |z|>|a|

Properties of Z transform:

• Linearity ax1[n] +bx2[n] z aX1(z) + bX2(z)

• Time shifting x[n-n0] z z-n0X(z)

• Scaling in the z domain z0nx[n] z X()

Properties :

• Time Reversal X[-n] z X() ROC =

• Conjugation x*[n] z X*(z*)

• Convolution x1[n] x2[n] z X1(z)X2(z)

Analysis of LTI Systems using Z- Transform

Analysis of LTI Systems using Z- Transform

• From the convolution property Y(z) = H(z) X(z) Where

Y(z)= z-transform of system output. H(z)= z-transform of impulse response. X(z) = z-transform of system input .

Stability and Causality

• Causality– A discrete time LTI system is causal if and only if

the ROC of its system function is the exterior of the circle, including infinity.

• Stability– The LTI system is stable if and only if the ROC of

the system function H(z) includes the unit circle, |Z| = 1

Stability and Causality for LTI systemwith Rational system Function

• Causality– The ROC is the exterior of the outermost pole.– With H(z) expressed as a ratio of polynomials in z,

the order of numerator cannot be greater than the order of denominator.

• Stability– If it is a causal system, it will be stable if and only if

all the poles of H(z) lie inside the unit circle – i.e. they must all have magnitude smaller than 1.

– It is possible for a system to be stable but not casual.

THANK YOU

- Pranvendra Champawat- 08010824- p.champawat@iitg.ernet.in