DSP#1
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 nonperiodic, continuous signal.
### 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 nonperiodic, continous signal as a sum of sinusoids?
We can manipulate the nonperiodic 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

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

Duality
 pulse <> sinc function
 delta <> constant

Parseval
 Convolution
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)

Compute X(w) by FT

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

Compute y(t) by inverse FT
Appendix
Derive \(a_k\) for FS
Fourier series of a pulse train
FT as FS where period goes to infinity
Deriving FT for various functions
References
 DSP lectures by Rich Radke
 Fourier series by 3Blue1Brown