3. The Frequency Domain

In the section on LTI systems we have seen that complex expontials are the eigenfunctions of LTI systems. Let \(x(t)=\exp(-j\omega t)\) then the output of an LTI system with impulse response \(h(t)\) is given as \(y(t)=H(\omega)x(t)\).

where \(H(\omega)\) is completely determined by the impulse response \(h(t)\):

\[H(\omega) = \int_{-\infty}^{\infty} h(t) e^{-j \omega t} dt\]

What would happen if the eigenfunctions would form a basis of signal (function) space, i.e. what if we could write any function \(x(t)\) as a linear combination of complex exponential functions? Because we are dealing with a linear system the response to \(x(t)\) would be the linear combination of the responses of all transformed complex exponentials. So knowing what an LTI system does for any frequency \(\omega\) enables us to calculate what the system does for an arbitrary signal.

This in a nutshell is what the frequency domain analysis of signals and systems is all about.

In this chapter we will look at Fourier Series and Fourier transforms both for CT and DT signals. All the four combinations are closely related and all are dealt with in this chapter.

The table below states all four versions of “Fourier Analysis”. For now you need not understand what this all means. We will start this chapter with the second row and only then look at discrete time signals and systems.

	Periodic	Non-periodioc
Discrete Time	DT Fourier Series \[x[n] = \sum_{k=0}^{N-1} a_k e^{j k \frac{2\pi}{N} n}\] \[a_k = \frac{1}{N}\sum_{n=0}^{N-1} x[n] e^{-j k \frac{2\pi}{N} n}\]	DT Fourier Transform \[x[n] = \frac{1}{2\pi} \int_{\langle 2\pi \rangle} X(\Omega) e^{j\Omega n} d\Omega\] \[X(\Omega) = \sum_{n=-\infty}^{\infty} x[n]e^{-j \Omega n}\]
Continuous Time	CT Fourier Series \[x(t) = \sum_{k=-\infty}^{\infty} a_k e^{j k \omega_0 t}\] \[a_k = \frac{1}{T_0}\int_{\langle T_0 \rangle} x(t) e^{-j k \omega_0 t} dt\]	CT Fourier Transform \[x(t) = \frac{1}{2\pi} \int_{-\infty}^{\infty} X(\omega) e^{j\omega t} d\omega\] \[X(\omega) = \int_{-\infty}^{\infty} x(t) e^{-j \omega t} dt\]

• 3.1. Continuous Time Fourier Series
  • 3.1.1. The Complex Exponential Functions
  • 3.1.2. The Fourier Series
  • 3.1.3. Properties of the CT Fourier Series
    • 3.1.3.1. Real Valued Signal
    • 3.1.3.2. Even Signal
    • 3.1.3.3. Odd Signal
    • 3.1.3.4. Differentiation
    • 3.1.3.5. Convolution
    • 3.1.3.6. Fourier Series Examples
      • 3.1.3.6.1. Block Wave Function
      • 3.1.3.6.2. From Time Domain to Frequency Domain
      • 3.1.3.6.3. Triangular Signal
  • 3.1.4. Exercises
• 3.2. Continuous Time Fourier Transform
  • 3.2.1. The Fourier Transform
  • 3.2.2. Properties of the CT Fourier Transform
    • 3.2.2.1. Real Signals
    • 3.2.2.2. Even and Odd Signals
    • 3.2.2.3. Derivatives
    • 3.2.2.4. Pulse Function
    • 3.2.2.5. Time Shifts
    • 3.2.2.6. Complex Exponential
    • 3.2.2.7. Periodic Signal
    • 3.2.2.8. Pulse Train
    • 3.2.2.9. Convolution
    • 3.2.2.10. Duality
  • 3.2.3. Fourier Transform Pairs
    • 3.2.3.1. Complex Exponential \(\FTright\) Pulse
    • 3.2.3.2. Pulse \(\FTright\) Complex Exponential
  • 3.2.4. Bode Plots
  • 3.2.5. Exercises
• 3.3. The Discrete Time Fourier Transform
  • 3.3.1. DTFT and IDTFT
  • 3.3.2. Properties of the DTFT
  • 3.3.3. Fourier Transform Pairs
  • 3.3.4. Introducing Time
  • 3.3.5. Exercises
• 3.4. Discrete Time Fourier Series
  • 3.4.1. Synthesis and Analysis Equations
  • 3.4.2. Properties of Discrete Time Fourier Series
  • 3.4.3. Time and Frequency
  • 3.4.4. DTFS in Numpy
  • 3.4.5. DTFS in Linear Algebra Disguise
  • 3.4.6. The Fast Fourier Transform
  • 3.4.7. Exercises