Notes from lectures 1 - 6

0. Key ideas

  • The goal is to represent any continous signals as a sum of sinusoids (aka. complex exponentials).
  • Fourier series is for periodic, continuous signal, while fourier transform is for non-periodic, continuous signal.

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### 1. Fourier series

All periodic, continuous signals can be represented as a sum of sinusoids.

\[x(t) = \sum_{k=-\infty}^{\infty}a_k e^{jk\omega_0t}\] \[a_k = \frac{1}{T} \int_0^T x(t)e^{-jk\omega_0t}dt\]

\(a_k\) is a coefficient, which represents the magnitude of the \(k\)-th sinusoid with frequency \(k\omega_0\). These coefficients are complex values.

1.1 What if input signal \(x\) is real?

Then, \(a_k\) is symmetric. \(a_k = a_{-k}*\).

If the signal is real, we can represent it only with cosines, since cosines are the real part of the complex exponentials.

\[x(t) = a_0 + \sum 2A_k cos(\theta_k + k\omega_0t)\]

Where \(A_k\) is the amplitude and \(\theta_k\) is the phase shift value of cosines.

1.2 What is the fourier series of a pulse train?

\(a_k\)s are sinc functions.

1.3 Properties of FS

  • Linearity
  • Time shifting
    • The \(a_k\) of the time shifted signal is a phase shifted version of the \(a_k\) of the original signal.
    • No magnitude changes.
  • Differentiation
  • Parseval’s
  • Convolution
    • Convolution in one domain is a multiplication in the other domain

1.4 Gibbs phenomenon

2. Fourier transform

Not all signals are periodic. How can we represent non-periodic, continous signal as a sum of sinusoids?

We can manipulate the non-periodic signal as having an infinitely long period and use FS to derive FT. As a result,

\[x(t) = \frac{1}{2\pi}\int_{-\infty}^{\infty} X(\omega)e^{j\omega t}d\omega\]

where,

\[X(\omega) = \int_{-\infty}^{\infty} x(t)e^{-j\omega_0t}dt\]

These are called synthesis and analysis equations, respectively.

2.2 Properties of FT

  • Linearity
  • Time shift
  • Symmetry
    • Even(\(x(t)\)) <-> Re(\(X(\omega)\))
    • Odd(\(x(t)\)) <-> \(j\)Im(\(X(\omega)\))
  • Differentiation/integration
\[x'(t) \Leftrightarrow j\omega X(\omega)\] \[\int_{-\infty}^t x(\tau)d\tau \Leftrightarrow \frac{1}{j\omega}X(\omega) + \pi X(0)\delta(\omega)\int_{-\infty}^t x(\tau)d\tau \Leftrightarrow \frac{1}{j\omega}X(\omega) + \pi X(0)\delta(\omega)\]

  • Time scaling \(x(at) \Leftrightarrow \frac{1}{\vert a \vert} X(\frac{\omega}{a})\)

  • Duality

    • pulse <-> sinc function
    • delta <-> constant

  • Parseval

\[\int_{-\infty}^{\infty} \lvert{x(t)}\rvert^2dt = \frac{1}{2\pi}\int_{-\infty}^{\infty} \lvert{X(\omega)}\rvert^2 d\omega\]
  • Convolution
\[y(t) = x(t) * h(t)\] \[Y(\omega) = X(\omega)H(\omega)\]

3. Frequency response

Frequency response (H(w)) : Fourier transform of the impulse response. Aka. filter.

⇒ Information about how each frequency will change after going through the LTI system

Given x(t) and H(w), we can calculate the output signal y(t)

  1. Compute X(w) by FT

  2. Compute Y(w) = H(w)X(w)

  3. Compute y(t) by inverse FT

Appendix

Derive \(a_k\) for FS

Engineering Math-28

Engineering Math-29

Fourier series of a pulse train

Engineering Math-30

FT as FS where period goes to infinity

Engineering Math-31

Engineering Math-32

Deriving FT for various functions

Engineering Math-33

Engineering Math-34

Engineering Math-35

Table of fourier transforms

References