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

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

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: http://users.ece.gatech.edu/mcclella/matlabGUIs/ Analogy with multiplication of two polynomials Linear algebra and vector dot product Prior lecture Today’s lecture

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)

A Brief Detour of Linear Algebra and Dot Products 𝑦 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

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?

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

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