Contents1 Signals 111.1 Basic Signals1.1.1 Unit-impulse Signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.1.2 Unit-step Signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.1.3 Unit-ramp Signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.1.4 Sinusoids and Complex Exponentials . . . . . . . . . . . . . . . . . . . . . . . 141.2 Classification of Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191.2.1 Continuous, Discrete, and Digital Signals . . . . . . . . . . . . . . . . . . . . 191.2.2 Periodic and Aperiodic Signals . . . . . . . . . . . . . . . . . . . . . . . . . . 191.2.3 Even- and Odd-symmetric Signals . . . . . . . . . . . . . . . . . . . . . . . . 201.2.4 Energy and Power Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221.2.5 Deterministic and Random Signals . . . . . . . . . . . . . . . . . . . . . . . . 241.2.6 Causal and Noncausal Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . 241.3 Signal Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241.3.1 Time Shifting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241.3.2 Time Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251.4 Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292 The Discrete Fourier Transform 332.1 The Exponential Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.2 The Complex Exponential Function . . . . . . . . . . . . . . . . . . . . . . . . . . . 342.2.1 Euler's Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342.2.2 Real Sinusoid in terms of Complex Exponentials . . . . . . . . . . . . . . . . 352.3 The DFT and the IDFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352.3.1 The DFT and the IDFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382.3.2 The Criterion of Approximation . . . . . . . . . . . . . . . . . . . . . . . . . 392.3.3 The Matrix form of the DFT and IDFT . . . . . . . . . . . . . . . . . . . . . 412.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462.4.1 Fourier Boundary Descriptor . . . . . . . . . . . . . . . . . . . . . . . . . . . 472.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493 Properties of the DFT 533.1 Linearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533.2 Periodicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533.3 Circular Time Shifting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543.4 Circular Frequency Shifting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543.5 Circular Time-reversal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553.6 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553.7 Transform of Complex Conjuagtes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563.8 Circular Convolution and Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . 563.8.1 Circular convolution of Time-domain Sequences . . . . . . . . . . . . . . . . . 5673.8.2 Circular Convolution of Frequency-domain Sequences . .
Dr. D. Sundararajan holds a B.E. in Electrical Engineering from Madras University and an M.Tech. in Electrical Engineering from the Indian Institute of Technology Chennai (IIT Chennai). He obtained his Ph.D. in Electrical Engineering at Concordia University, Montreal, Canada in 1988. As the principal inventor of the latest family of discrete Fourier transform (DFT) algorithms, he holds three patents (granted by the US, Canada and Britain). Further, he has published several papers in IEEE Transactions and in the Proceedings of the IEEE Conference, and he is the author of five books. He has taught undergraduate and graduate classes in digital signal processing, digital image processing, engineering mathematics, programming, operating systems and digital logic design at Concordia University, Canada, Nanyang Technological University, Singapore, and Adhiyamaan College of Engineering, India. He has also conducted workshops on Digital image processing, MATLAB and LaTeX.
Overthe course of his engineering career, he has held positions at the National Aerospace Laboratory, Bangalore, and the National Physical Laboratory, New Delhi, where he worked on the design of digital and analog signal processing systems.
