1. CT-321 Digital Signal Processing
Yash Vasavada
Autumn 2016
DA-IICT
Lecture 6
LSI Systems and DTFT
16th August 2016

2. Review and Preview Review of past lecture: Preview of this lecture:
Different ways of understanding the convolution operation
Preview of this lecture:
Convolution Operation
Relation to DTFT
Properties of DTFT and LSI Systems
Reading Assignment
OS, 3rd Edition: Sections 1.2 to 1.4, and Sections 1.6 to 1.9
Note: available in the library
PM: Sections 2.1 to 2.3

3. Convolution Operation
Output of an LSI system can be written as a function of its input 𝑥 𝑛 and impulse response ℎ(𝑛):
𝑦 𝑛 = 𝑘=−∞ ∞ 𝑥 𝑘 ℎ 𝑛−𝑘
This formula represents the convolution operation between two D-T sequences 𝑥(𝑛) and ℎ(𝑛)
There are several ways to understand the convolution operation.
Use the principles of superposition and homogeneity
Flipping the impulse response and slide it over the input 𝑥(𝑛): demo on next several slides
From:
Analogy with multiplication of two polynomials
Linear algebra and vector dot product
Prior lecture
Today’s lecture

4. Polynomial Multiplication
Consider two polynomials:
𝑝 𝑥 = 𝑝 0 + 𝑝 1 𝑥+ 𝑝 2 𝑥 2 +…+ 𝑝 𝑁 𝑥 𝑁
𝑞 𝑥 = 𝑞 0 + 𝑞 1 𝑥+ 𝑞 2 𝑥 2 +…+ 𝑞 𝑀 𝑥 𝑀
Product of these two polynomials is given as
𝑟 𝑥 =𝑝 𝑥 𝑞 𝑥 = 𝑟 0 + 𝑟 1 𝑥+ 𝑟 2 𝑥 2 +…+ 𝑟 𝑁+𝑀 𝑥 𝑁+𝑀
Coefficients of this product polynomial can be calculated by convolution of the coefficients of 𝑝(𝑥) and 𝑞(𝑥)
Define: 𝑝 𝑛 = 𝑝 𝑛 , 𝑞 𝑛 = 𝑞 𝑛 and 𝑟 𝑛 = 𝑟 𝑛
𝑟 𝑛 =𝑝 𝑛 ∗𝑞(𝑛)
An indication of a proof showing why convolution in one domain (say, time domain) is equivalent to multiplication in another frequency domain
Evaluate 𝑝(𝑥) and 𝑞(𝑥) and 𝑥=𝑢 (equivalent to taking DTFT at frequency 𝑓) and take product to get 𝑟(𝑢), or
Convolve 𝑝(𝑛) and 𝑞(𝑛) (equivalent to passing an input 𝑝(𝑛) through an LSI system with impulse response 𝑞(𝑛)) to get 𝑟(𝑛) and use that to evaluate 𝑟(𝑥=𝑢)

5. A Brief Detour of Linear Algebra and Dot Products𝑦
Consider a point 𝐴 on a unit circle in 2D plane
This can be represented by a vector 𝑨= 𝐴 1 , 𝐴 2 𝑇
Convention is to highlight (bold-face) the notation of a vector, which is usually a column of numbers.
Notation 𝑇 stands for transpose, and it is used to convert a row vector into a column vector and vice versa
Suppose this vector 𝑨 makes an angle of 𝜃 𝐴 with respect to 𝑋 axis
Therefore 𝐴 1 = cos 𝜃 𝐴 and 𝐴 2 = cos 90− 𝜃 𝐴 = sin 𝜃 𝐴
We consider another point 𝐵 and corresponding vector 𝐵=[ cos 𝜃 𝐵 , sin 𝜃 𝐵 ]
What is the value of 𝐴 1 𝐵 1 + 𝐴 2 𝐵 2 ?
Answer: cos 𝜃 𝐴 − 𝜃 𝐵 𝐵 𝐴
𝐴 2
𝜃 𝐵
𝜃 𝐴 𝑥
𝐴 1
Unit length circle in 2D plane
Angle between the two vectors

6. A Brief Detour of Linear Algebra and Dot Products
The dot product of two 𝑛×1 (real-valued) vectors 𝑨 = 𝐴 1 , 𝐴 2 , …, 𝐴 𝑛 𝑇 and 𝑩 = 𝐵 1 , 𝐵 2 , …, 𝐵 𝑛 𝑇 is defined as
Here, 𝑨 denotes the norm of vector 𝑨, which is effectively its length
Suppose vectors 𝑨 and 𝑩 are both unit length vectors. In this case, the dot product 𝑨∙𝑩 is simply cos 𝜃 , i.e., the cosine of the angle between the two vectors
Generalization of the prior slide for which 𝑛=2
This has a nice visualization: when the unit-norm vectors 𝑨 and 𝑩 are in a good alignment, angle 𝜃 between them is near zero and the dot product approaches the maximum value of unity
When does the dot product between vectors 𝑨 and 𝑩 become zero?

7. Convolution Operation as Sequential Dot Products
Output 𝑦= 𝑆𝑦𝑠𝑡𝑒𝑚 𝕋 or matrix 𝐻 × Input 𝑥
Define 𝒉= ℎ 𝑀 , ℎ 𝑀−1 ,…, ℎ 1 , ℎ 0 𝑇 as a system vector (flipped impulse response), and 𝒙 𝑛 = 𝑥 𝑛 , 𝑥 𝑛−1 ,…, 𝑥 𝑛−𝑀+1 , 𝑥 𝑛−𝑀 𝑇 as a vector of input signal samples from samples 𝑛−𝑀 to 𝑛
With this, 𝑛 𝑡ℎ output sample is 𝑦 𝑛 = 𝒉 𝑇 𝒙 𝑛 , i.e., it’s a dot product of system vector 𝒉 with 𝒙 𝑛
𝑦 𝑛 will have a high value if 𝒙 𝑛 is well aligned with 𝒉, and will have a low value otherwise

8. An Aside: Why the Word “Linear”?
Any system 𝕋 that satisfies the properties of superposition and scaling by a constant is called a linear system
Reason for the word “linear” is that such systems allow a matrix viewpoint
Output 𝑦(𝑛) of such systems can be represented as a vector 𝒚=𝑯𝒙
As a course in elementary linear algebra shows, it’s only lines and their generalizations (planes, etc.) that also can be described in such a way.
A typical equation of a line: 𝑦=𝑚𝑥+𝑐, where 𝑚 is the slope and 𝑐 is the abscissa
This can be written as a matrix product 𝒚=𝑪𝒃, where 𝑦= 𝑦 1 , 𝑦 2 ,…, 𝑦 𝑀 𝑇 is a vector of the Y- coordinates of 𝑀 points along this line, 𝑪= 𝑥 1 1 ⋮ ⋮ 𝑥 𝑀 1 , and 𝒃= 𝑚 𝑐
Nonlinear functions such as powers, logarithms, exponentials, etc. have non-constant slope and they do not have such a matrix product representation