FOUNDATION
Learn Signals and Systems from the basics
Short chapters, in reading order. Each one explains an idea from first principles and leads into the topic notes.
The chapters, in order
CHAPTER 1
Complex numbers, Euler's formula and the complex exponential
This chapter builds the complex number, Euler's formula and the complex exponential e^st from first principles, then adds roots of unity and the geometric series.
Leads into the GATE topics: Signals basics · Fourier series · DTFT & DFT · z-transform
CHAPTER 2
The integrals, sums and limits that keep coming back
This chapter works through the integrals and sums that signal transforms are built from: exponentials over half-lines, energy and power, swapping a sum and an integral, and the sinc integral.
Read first: Chapter 1, Complex exponential
Leads into the GATE topics: Signals basics · Fourier transform · Laplace · z-transform
CHAPTER 3
What a system is, and why linear and time-invariant matter
This chapter treats a system as a rule from input signal to output signal, tests three models for linearity and time invariance, and derives convolution from those two properties.
Read first: Chapter 1, Complex exponential
Leads into the GATE topics: LTI & convolution
CHAPTER 4
Convolution, slowly
This chapter works convolution out by hand: flip and slide with every interval written down, discrete convolution as polynomial multiplication, and the three laws read as cascaded and parallel systems.
Read first: Chapter 3, Systems, LTI, convolution
Leads into the GATE topics: LTI & convolution · z-transform
CHAPTER 5
Why sinusoids: eigenfunctions and the frequency idea
This chapter shows, from convolution, why a complex exponential leaves a linear time-invariant system changed only in gain and phase.
Read first: Chapter 1, Complex exponential, Chapter 4, Convolution, slowly
Leads into the GATE topics: Fourier series · Fourier transform
CHAPTER 6
From series to transform: one family, five names
This chapter shows that the Fourier series, Fourier transform, Laplace, z, DTFT and DFT are one idea with different freedoms.
Read first: Chapter 2, Integrals and limits, Chapter 5, Why sinusoids
Leads into the GATE topics: Fourier transform · Laplace · z-transform · DTFT & DFT
CHAPTER 7
Sampling from first principles
Sampling at a steady rate copies a spectrum at every multiple of the sampling frequency.
Read first: Chapter 6, One transform family
Leads into the GATE topics: Sampling · DTFT & DFT
CHAPTER 8
Reading a system from its poles and zeros
This chapter shows where poles come from, why their position decides stability, and how lengths and angles on the s-plane and z-plane give a response's magnitude and phase.
Read first: Chapter 6, One transform family
Leads into the GATE topics: Laplace · z-transform · FIR & IIR filters · Nyquist (Control Systems)