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Introduction to Signals

The World of Signals Signals Are Everywhere What is Digital Signal Processing?Applications of DSP

Basic Signals and Operations

How to Generate a Basic Sinusoid A Unit Impulse Signal Operations on Signals How to Shift a Signal in Time Domain The Relation between Time Shift and Phase Circular Shift Correlation between Signals Activity 1

Complex Signals and the Frequency Domain

Where Do Complex Numbers Come From?Simplify Complex Operations Phase Rotations Complex Sinusoids: The Doorway to DSP Negative Frequencies A Real Sinusoid: The DNA of Signals Frequency Domain Building Blocks of All Signals Spectrum and Bandwidth Spectral Shift in Wireless Communication Quiz 1

Sampling a Signal

From Continuous Time to DiscreteTime Why Do Spectral Aliases Arise?Aliasing and the Sampling Theorem Sample Rate and Discrete Frequency The Idea of Orthogonality Discrete Frequency Axis Number Representation Quiz 2

The Discrete Fourier Transform

DFT and the Mysteries of the Universe DFT Definition The Spectrum of the Unit Impulse Reading Amplitude and Phase Plots Periodicity in Frequency and Time Domains A Rectangular Signal Main Lobe Peak: Width and Zero Crossings DFT Phase DFT Leakage Fast Fourier Transform The Inverse Discrete Fourier Transform Quiz 3

Fourier Transform Properties

Spectrum After the Time Shift I

Spectrum After the Time Shift II

The Significance of a Linear Phase

Frequency Shifting and Modulation

Parseval's Theorem

Symmetry Properties

Digital Systems

What Is a System?Linear Systems Time Invariance System Impulse Response System Frequency Response Decomposition of a Signal into Scaled Unit Impulses Response of an LTI System How to Implement Convolution Simplified Convolution Convolution of Complex Signals The Role of Complex Sinusoids in System Characterization Circular Convolution Convolution and the Frequency Domain Filter Classifications in the Time Domain Filter Classifications in the Frequency Domain Activity 2

Putting It All Together: An OFDM Modem

Introduction to Orthogonal Frequency Division Multiplexing Digital Modulation IFFT and FFT Wireless Channel The Cyclic Prefix Equalization Digital Demodulation Quiz 5

Course Assessment

Demodulating an OFDM Signal Stream

The Last Words

Bibliography

Fundamentals of Digital Signal Processing

Parseval's Theorem

Discover the relation between signal energy in the time and frequency domains.

We'll cover the following...

Derivation
Intuition

The energy of a length- $N$ discrete-time signal is defined as:

E_x = \sum_{n=0}^{N-1} |x[n]|^2