Lecture 12: The z-Transform and LTI systems

Lecture 12:The z-Transform and LTI systems Instructor: Dr. Ghazi Al Sukkar Dept. of Electrical Engineering The University of Jordan Email: ghazi.alsukkar@ju.edu.jo

Outline System function System properties: Causality Stability LCCDE systems representation Inverse System properties Unilateral z-Transform The Frequency Response

Inverse System • Recall that: • An LTI can be represented as a Constant-coefficient difference equation • Systems described as difference equations have system functions of the form we may assume • Given an LTI system H(z) the inverse system Hi(z) is given as • The cascade of a system and its inverse yields unity

Cont.. • If it exists, the frequency response of the inverse system is • Not all systems have an inverse: zeros cannot be inverted • Example: Ideal lowpass filter (there is no way to recover the frequency components above the cutoff frequency that are set to zero) • The inverse of rational system functions • The poles of are the zeros of and vice versa. • ROC of inverse has to overlap with ROC of original system

Examples: Inverse System • Example 1: Let’s find the inverse system of • The ROC of the inverse system is either • Only overlaps with original ROC • Example 2: Let’s find the inverse system of • Again two possible ROCs • This time both overlap with original ROC so both are valid • Two valid inverses for this system

Cont.. • If is causal with zeros, then the inverse system is causal if and only if the associated ROC for is such that • If the causalinverse system is required to be stable, then the ROC of must include the unit circle: i.e., all the zeros of must be inside the unit circle. • Hence, an LTI system is stable and causal and also has a stable and causal inverse system if and only if both the poles and the zerosof are inside the unit circle (called minimum phase systems).

The Unilateral z-Transform: • The one-sided z-transform of a sequence is defined as: • Then the sample shifting property is given by Let By letting in the summation:

Solve difference equations with nonzero initial conditions • Solve: Subject to these initial conditions: Example: Solve:

Cont..

Forms of the solutions: • Homogeneous and particular parts: • The homogeneous part is due to the system poles and the particular part is due to the input poles. • Transient and steady-state response: • The transient response is due to poles that are inside the unit circle, while the steady-state response is due to poles that are on the unit circle. • Note that when the poles are outside the unit circle, the response is termed an unbounded response.

Cont.. • Zero-input and zero-state responses:

Cont.. • Therefore, Matlab function: y=filter(b,a,x,xic) • Xic is an equivalent initial-condition input array • Xic = filtic(b,a,Y,X) • b and a are the filter coefficient array • Y and X are the initial-condition arrays from the initial conditions on y[n] and x[n], respectively, in the form

Cont.. • Final value theorem is used to find the steady state value of a function:

Quick Review of LTI Systems • LTI Systems are uniquely determined by their impulse response • We can write the input-output relation also in the z-domain • Or we can define an LTI system with its frequency response • H(ej) defines magnitude and phase change at each frequency • We can define a magnitude response (gain) • And a phase response (phase shift)

Ideal Low Pass Filter • Ideal low-pass filter

Ideal High-Pass Filter • Can be written in terms of a low-pass filter as

Signal distortion by LTI systems • A system has distortionless response if the input signal and the output signal satisfy the condition where and are constants. • In the Fourier domain, this condition amounts to • Therefore, the frequency response of a distortionless LTI system should satisfy • Conclusion: For a distortionless LTI system, the magnitude response must be a constant and the phase response must be a linear function of with slope , where is a relative delay of the output signal.

Cont.. • We say that a system introduces magnitude distortion if • LTI systems without magnitude distortion are known as allpasssystems. Allpasssystems have a “flat” magnitude response and their characteristics are completely determined by the phase response. • If the phase response is not a linear function of frequency, that is, the resulting distortion is known as phase or delaydistortion.

Phase delay • Recall that the phase response gives the phase shift experienced by each sinusoidal component of the input signal, i.e., • In the case when , we have where the quantity shows the time shift (in number of sampling intervals) experienced by each sinusoidal component of the input signal.

Cont.. • Therefore, sometimes it is more meaningful to use the phase delay defined by • Nonlinear phase responses may lead to severe shape alterations.

Example: Different phase delays Note that the constant phase shift in causes distortion because each frequency component is delayed by a different amount. In contrast, the linear-phase shift in does not cause any distortion because it results in a constant phase delay.

Phase Distortion and Delay • Remember the ideal delay system • In terms of magnitude and phase response • Delay distortion is generally acceptable form of distortion • Translates into a simple delay in time • Also called a linear phase response • Generally used as target phase response in system design • Ideal lowpass or highpass filters have zero phase response • Not implementable in practice

Ideal Low-Pass with Linear Phase • Delayed version of ideal impulse response • Filters high-frequency components and delays signal by nd • Linear-phase ideal lowpass filters is still not implementable

Unwrapped (Continuous) Phase • Phase is ambiguousWhen calculating the arctan(.) function on a computer • Values between - and + • Denoted in the book as ARG(.): Principal value (wrapped phase) • Any multiple of 2 would give the same result • Here r() is a +ve or –ve integer for any given value of  • defines unwrapped or continuous phase

Group delay • A convenient way to check the linearity of phase response is to use the groupdelay, defined as the negative of the slope of the unwrapped phase as follows: • Phase responses which are linear in frequency correspond to constant phase delay and constant group delay; both delays are identical, and each may be interpreted as time delay. • Note that both the linear-phase response and the generalized linear phase response have a constant group delay. • The name group delay comes because shows the delay of the “bundle” of frequency components about .

Cont.. • Example: Consider with and . The input consists of two consecutive narrowband pulses and with And is a Gaussian envelope.

Cont.. • The first pulse, which is centered at the passband of the filter, passes through with a group (aka envelope delay) of approximately 60 samples. The attenuation and smearing of the envelope is due to the magnitude distortion of the filter. • The second pulse is attenuated by more than 100 dB and it does not appear in the output.

Outline System function System properties: Causality Stability LCCDE systems representation Inverse System properties Unilateral z-Transform The Frequency Response of Rational System Function

Frequency Response of Rational System Functions • DTFT of a stable and LTI rational system function • Magnitude Response • Magnitude Squared

Log Magnitude Response • Log Magnitude in decibels (dB) • Example: • |H(ej)|=0.001 translates into –60dB gain or 60dB attenuation • |H(ej)|=1 translates into 0dB gain • |H(ej)|=0.5 translates into -6dB gain • Output of system

Phase Response • Phase response of a rational system function • Corresponding group delay • Here arg[.] represents the continuous (unwrapped) phase • Work it out to get • Hint:

Cont.. • Also

Frequency Response of a Single Zero or Pole • Let’s analyze the effect of a single term • If we represent it in dB • The phase term is written as • And the group delay obtained by differentiating the phase • Maximum and minimum value of magnitude

Cont.. • Some examples • One zero • Value of r • Value of • Magnitude response • Phase response • One pole • Magnitude response • Phase response • You can practice: multiple poles and zeros

Example: r=0.9, Radians

r=0.9, Radians

r=0.99,

r=0.7,

r=0.5,

r=0.3,

One pole • r=0.99 and 0.7,

r=0.9, , and

Vector Diagram in z-plane • A helpful method in inferring the frequency response characteristics. • Each pole and zero factor is represented by a vector in the z-plane from the pole or zero to a point on the unit circle. • For 1st order system function of the form • Since But Unit circle

Cont.. • Also since • The vector has minimum length when . Frequency response for a single zero with θ = π, r = 1, 0.9, 0.7, and 0.5.

Design of simple Filters • There is an intuitive strong dependence between the locations of poles and zeros and the frequency response of a system. • Thus, for example, to suppress a frequency component at , we should place a zero at angle on the unit circle. • Alternatively, to enhance or amplify a frequency component at , we should place a pole at angle close but inside the unit circle. • Complex poles or zeros should appear in complex conjugate pairs to assure that the system has real coefficients. • It can be shown that poles or zeros at the origin do not influence the magnitude response. However, a pole (or zero) at the origin adds (or subtracts) a linear phase to the phase response.

Lecture 12: The z-Transform and LTI systems