Signals and Systems for All Electrical Engineers

Aaron D. Lanterman, Jennifer E. Michaels, and Magnus B. Egerstedt

August 22, 2016 2 Contents

Change Log i

Preface iii 0.1 The DSP First Legacy ...... iii 0.2 Forward to the Past ...... iv 0.3 Putting Educational Eggs in Many Baskets – Not Just One ...... v 0.4 For All Electrical Engineers ...... v 0.5 Tough Choices ...... vi

1 What are Signals? 1 1.1 Convenient continuous-time signals ...... 1 1.1.1 Unit step functions ...... 1 1.1.2 Delta “functions” ...... 3 1.1.3 Calculus with Dirac deltas and unit steps ...... 4 1.2 Shifting, flipping and scaling continuous-time signals in time ...... 4 1.3 Under the hood: what professors don’t want to talk about ...... 5

2 What are Systems? 9 2.1 System properties ...... 10 2.1.1 Linearity ...... 10 2.1.2 Time-invariance ...... 11 2.1.3 Causality ...... 11 2.1.4 Examples of systems and their properties ...... 12 2.2 Concluding thoughts ...... 13 2.2.1 Linearity and time-invariance as approximations ...... 13 2.2.2 Contemplations on causality ...... 14 2.2.3 How these properties play out in practice in a typical “signals and systems” course . . 14

3 Why are LTI Systems so Important? 15 3.1 Review of convolution for discrete-time signals ...... 15 3.2 Convolution for continuous-time signals ...... 16 3.3 Review of frequency response of discrete-time systems ...... 16 3.4 Frequency response of continuous-time systems ...... 17 3.5 Connection to Fourier transforms ...... 18

3 4 CONTENTS

3.6 Finishing the picture ...... 19 3.7 A few observations ...... 19

4 More on Continuous-Time Convolution 21 4.1 The convolution integral ...... 21 4.2 Properties of convolution ...... 22 4.3 Convolution examples ...... 23 4.4 Some final comments ...... 28

5 Cross-Correlation and Matched Filtering 29 5.1 Cross-correlation properties ...... 30 5.2 Cross-correlation examples ...... 31 5.3 Matched filter implementation ...... 31 5.4 Delay estimation ...... 32 5.5 Causal concerns ...... 33 5.6 A caveat ...... 34 5.7 Under the hood: squared-error metrics and correlation processing ...... 34

6 Review of 37 6.1 Fourier synthesis sum and analysis integral ...... 37 6.2 System response to a periodic signal ...... 38 6.3 Properties of Fourier series ...... 38 6.4 Fourier series of a symmetric “square wave” ...... 39 6.4.1 Lowpass filtering the square wave ...... 40 6.5 What makes Fourier series tick? ...... 41 6.6 Under the hood ...... 43

7 Fourier Transforms 45 7.1 Motivation ...... 45 7.2 A key observation ...... 46 7.3 Your first : decaying exponential ...... 47 7.3.1 Frequency response example ...... 47 7.4 Your first Fourier transform property: time shift ...... 48 7.5 Your second Fourier transform: delta functions ...... 48 7.5.1 Sanity check ...... 49 7.6 Rectangular boxcar functions ...... 49 7.6.1 Fourier transform of a symmetric rectangular boxcar ...... 49 7.6.2 Inverse Fourier transform of single symmetric boxcar ...... 50 7.6.3 Observations about our boxcar examples ...... 50 7.7 Fourier transforms of deltas and sinusoids ...... 51 7.8 Fourier transform of periodic signals ...... 52

8 Modulation 53 8.1 Fourier view of filtering ...... 53 8.1.1 Filtering by an ideal lowpass filter ...... 54 8.2 Modulation property of Fourier transforms ...... 54 CONTENTS 5

8.2.1 Modulation by a complex sinusoid ...... 55 8.3 Double Side Band Amplitude Modulation ...... 55 8.3.1 DSBAM transmission ...... 55 8.3.2 DSBAM reception ...... 56 8.3.3 Practical matters ...... 57 8.4 Baseband representations of bandlimited signals ...... 58

9 Sampling and Periodicity 59 9.1 Sampling time-domain signals ...... 59 9.1.1 A Warm-Up Question ...... 59 9.1.2 Sampling: from ECE2026 to ECE3084 ...... 59 9.1.3 A mathematical model for sampling ...... 60 9.1.4 Practical reconstruction from samples ...... 63 9.2 Deriving the DTFT and IDTFT fro