Digital Signal and Image Processing Lecture Notes PDF

Digital Signal and Image Processing Lecture 7 – Digital Functions Dr. T. Akilan [email protected] © Dr. T. Akilan 1 Welcome back National Conference on Undergraduate Research (NCU...

Digital Signal and Image Processing Lecture 7 – Digital Functions Dr. T. Akilan [email protected] © Dr. T. Akilan 1 Welcome back National Conference on Undergraduate Research (NCUR) - 2024 o Nov. 8, 2023: Volunteer [Abstract Review] Call Closes o Dec. 8, 2023: Submission Portal Closes Assignment 1: Due date - Oct. 6 © Dr. T. Akilan 2  This Session Recap Digital Functions o Impulse o Step o Ramp o Power, Exponential, Sine Notation for Digital Signals Composite Functions Discussion on assignment # 1 (if time permits) Linear, Time-Invariant (LTI), Causal Systems (next week) Pop Quizzes for Recap © Dr. T. Akilan 3  Recap – Pop Quiz # 3: Causality of a Signal Answer the following questions: oWhat is a signal and what information does it convey? oLabel the following signals as causal, non-causal, or anti-causal sequences. Amplitude Amplitude Amplitude Sample ID Sample ID Sample ID CT vs DT → © Dr. T. Akilan 4  Recap Cont. – Pop Quiz # 6: CT vs. DT Signals What is a continuous signal? o Describe its characteristics. o Provide a few examples of continuous-time (CT) signals found in nature. Differentiate discrete-time (DT) signals and digital signals. CT Signal DT Signal List examples of CT signals in nature: Examples of DT signals in nature: o _________________________________ o DNA base sequence o _________________________________ o Number of students in a class o _________________________________ o Population of the nth generation of certain species Digital Functions © Dr. T. Akilan 5 Digital Functions – Unit-Impulse Function The unit impulse function or Plotting 𝛿 Function unit sample sequence is defined as the sequence with % set the sample range. k_min = -5; values 𝜹 𝒏. k_max = 10; Unit Impulse Sequence k = k_min:k_max; % x will be true when k = 0; otherwise, false. 𝟎 𝒏≠𝟎 x = (k==0); 𝜹𝒏 =ቊ 𝟏 𝒏=𝟎 % plotting stem(k, x, 'MarkerSize',8, 'LineWidth',2) xlabel('\fontsize{14} k') ylabel('\fontsize{14} \delta[k]') title('\fontsize{16} Unit impulse sequence') sample range Scaled Unit-impulse → © Dr. T. Akilan 6  Unit-Impulse Function Cont. Unit Impulse Sequence The unit impulse function, 𝛿[𝑛] has an amplitude of zero at all samples except 𝟎 𝒏≠𝟎 at 𝑛 = 0, where it has the value 1. 𝜹𝒏 =ቊ 𝟏 𝒏=𝟎 Every digital signal can be written as a sum of several impulse functions, using the amplitude at each sample. Determine the values of the following: a) 𝛿 = ________, b) 𝛿 = ________, c) 𝛿[−2] = ________ Amplitude Reversal → © Dr. T. Akilan 7  Scaled Unit-Impulse Function Any impulse signal can be a Example # 1: Amplitude Scaling scaled or time variant 𝑥[𝑛] = 4𝛿[𝑛] (shifted ver.) of the unit impulse (δ) function. 𝑥 𝑛 = 4𝛿[𝑛] Recall: A digital signal 𝑥 is given the notation 𝒙[𝒏] to indicate 𝛿[𝑛] that it has values only where the samples are taken at a finite interval: o 𝒏 - sample’s index, taken at time 𝑡 = 𝑛𝑇𝑆, where 𝑇𝑆 is the sampling period. Amplitude Reversal → © Dr. T. Akilan 8  Scaled Unit-Impulse Function - Amplitude Reversal Example # 2: Amplitude Scaling and Reversal 𝑥[𝑛] = −𝛿[𝑛] 𝑥[𝑛] = −2𝛿[𝑛] Shifted Unit-impulse: Time Shifting → © Dr. T. Akilan 9  Shifted Unit-Impulse Function – Time Shifting 𝑥[𝑛] = 𝛿[𝒏 – 𝒏𝟎] 𝑜𝑟 𝛿[𝒏 + 𝒏𝟎] δ[n] shifting factor Example # 3: o 𝑥[𝑛] = 𝛿[𝑛 − 2] time delay Computing 𝑋[𝑛] = 𝛿[𝑛 − 2] ⋯ -3 -2 –1 0 1 2 3 4 ⋯ ⋮ 𝑋[−2] = 𝛿[−2 − 2] = 𝛿[−4] = 0 𝑋[−1] = 𝛿[−1 − 2] = 𝛿[−3] = 0 𝑋 = 𝛿[0 − 2] = 𝛿[−2] = 0 o 𝑥[𝑛] = 𝛿[𝑛 + 2] 𝑋 = 𝛿[1 − 2] = 𝛿[−1] = 0 time 𝑋 = 𝛿[2 − 2] = 𝛿 = 1 advancement ⋯ -3 -2 –1 0 1 2 3 4 ⋯ 𝑋 = 𝛿[3 − 2] = 𝛿 = 0 ⋮ Exercise # 1 → © Dr. T. Akilan 10  Shifted Unit-Impulse Function – Exercise #1 Sketch the signal 𝑥[𝑛] = 𝛿[𝑛 − 3] 𝑥 𝑛 = 0 for n≠3 𝑥 3 =1 Exercise # 2 → © Dr. T. Akilan 11 Composite Function – Exercise #2 Write a function to describe the signal shown in the figure. 𝛿[𝑛] 𝛿[𝑛 − 2] 𝛿[𝑛 − 1] 𝛿[𝑛 − 3] 𝑥[𝑛] = 𝛿[𝑛] + 𝛿[𝑛 − 1] + 𝛿[𝑛 − 2] + 𝛿[𝑛 − 3]  Composite function Exercise # 3 → © Dr. T. Akilan 12 Composite Function – Exercise #3 Write a function to describe the signal in the figure. 4𝛿[𝑛] 3𝛿[𝑛 − 2] −1𝛿[𝑛 − 3] −2𝛿[𝑛 − 1] 𝑥[𝑛] = 4𝛿[𝑛] − 2𝛿[𝑛 − 1] + 3𝛿[𝑛 − 2] − 𝛿[𝑛 − 3]  Composite function Pop quiz # 7 → © Dr. T. Akilan 13 Pop Quiz # 7 Sketch the sequence of the following function, 𝒙 𝒏 = 𝜹 𝒏 + 𝟏 + 𝟎. 𝟓𝜹 𝒏 − 𝟏 + 𝟐𝜹 𝒏 − 𝟐. Time Scaling: Compression → © Dr. T. Akilan 14 Time Scaling – Example # 1 x[n] 𝑦[𝑛] = 𝑥[𝑎 × 𝑛] o a > 1 – compression in time by factor 𝑎 o 0