����PRESENTATION ON �LTI SYSTEM

• BRANCH-E & TC ENGG
• SUBJECT- DIGITAL SIGNAL PROCESSING
• CHAPTER – 3 – THE Z TRANSFORM & ITS APPLICATION TO THE ANALYSIS OF LTI SYSTEM
• TOPIC- DISCRETE TIME SIGNAL
• SEM-6^TH
• FACULTY – Er. ARADHANA DAS (Sr. LECTURER E & TC ENGG DEPARTMENT)
• AY-2021-2022, SUMMER-2022

​

• Objectives:�Representation of DT Signals�Response of DT LTI Systems�Convolution�Examples�Properties
• Resources:�MIT 6.003: Lecture 3�Wiki: Convolution�CNX: Discrete-Time Convolution�JHU: Convolution�ISIP: Convolution Java Applet

DIGITAL SIGNALS

ECE 8443 – Pattern Recognition

EE 3512 – Signals: Continuous and Discrete

• Are there sets of “basic” signals, x_k[n], such that:
• We can represent any signal as a linear combination (e.g, weighted sum) of these building blocks? (Hint: Recall Fourier Series.)
• The response of an LTI system to these basic signals is easy to compute and provides significant insight.
• For LTI Systems (CT or DT) there are two natural choices for these building blocks:

• Later we will learn that there are many families of such functions: sinusoids, exponentials, and even data-dependent functions. The latter are extremely useful in compression and pattern recognition applications.

Exploiting Superposition and Time-Invariance

• DT Systems:�(unit pulse)

• CT Systems:�(impulse)

EE 3512: Lecture 14, Slide 2

Representation of DT Signals Using Unit Pulses

EE 3512: Lecture 14, Slide 3

Response of a DT LTI Systems – Convolution

• Define the unit pulse response, h[n], as the response of a DT LTI system to a unit pulse function, δ[n].
• Using the principle of time-invariance:
• Using the principle of linearity:
• Comments:
• Recall that linearity implies the weighted sum of input signals will produce a similar weighted sum of output signals.
• Each unit pulse function, δ[n-k], produces a corresponding time-delayed version of the system impulse response function (h[n-k]).
• The summation is referred to as the convolution sum.
• The symbol “*” is used to denote the convolution operation.

convolution sum

convolution operator

EE 3512: Lecture 14, Slide 4

LTI Systems and Impulse Response

• The output of any DT LTI is a convolution of the input signal with the unit pulse response:
• Any DT LTI system is completely characterized by its unit pulse response.
• Convolution has a simple graphical interpretation:

EE 3512: Lecture 14, Slide 5

Visualizing Convolution

• There are four basic steps to the calculation:
• The operation has a simple graphical interpretation:

EE 3512: Lecture 14, Slide 6

Calculating Successive Values

• We can calculate each output point by shifting the unit pulse response one sample at a time:

• y[n] = 0 for n < ???

y[-1] =

y[0] =

y[1] =

…

y[n] = 0 for n > ???

• Can we generalize this result?

EE 3512: Lecture 14, Slide 7

Graphical Convolution

1

-1

2

1

-1

-1

k = -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9

EE 3512: Lecture 14, Slide 8

Graphical Convolution (Cont.)

1

-1

2

1

-1

-1

k = -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9

EE 3512: Lecture 14, Slide 9

Graphical Convolution (Cont.)

• Observations:
• y[n] = 0 for n > 4
• If we define the duration of h[n] as the difference in time from the first nonzero sample to the last nonzero sample, the duration of h[n], L_h, is�4 samples.
• Similarly, L_x = 3.
• The duration of y[n] is: L_y = L_x + L_h – 1. This is a good sanity check.
• The fact that the output has a duration longer than the input indicates that convolution often acts like a low pass filter and smoothes the signal.

EE 3512: Lecture 14, Slide 10

Examples of DT Convolution

• Example: unit-pulse

• Example: delayed unit-pulse

• Example: unit step

• Example: integration

EE 3512: Lecture 14, Slide 11

Properties of Convolution

• Commutative:

• Implications

• Distributive:

• Associative:

EE 3512: Lecture 14, Slide 12

Useful Properties of (DT) LTI Systems

• Causality:

• Stability:

Bounded Input ↔ Bounded Output

Sufficient Condition:

Necessary Condition:

EE 3512: Lecture 14, Slide 13

• We introduced a method for computing the output of a discrete-time (DT) linear time-invariant (LTI) system known as convolution.
• We demonstrated how this operation can be performed analytically and graphically.
• We discussed three important properties: commutative, associative and distributive.
• Question: can we determine key properties of a system, such as causality and stability, by examining the system impulse response?
• There are several interactive tools available that demonstrate graphical convolution: ISIP: Convolution Java Applet.

Summary

EE 3512: Lecture 14, Slide 14

Thank You