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 non-periodic, 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 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
-
Time scaling \(x(at) \Leftrightarrow \frac{1}{\vert a \vert} X(\frac{\omega}{a})\)
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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)
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Compute X(w) by FT
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Compute Y(w) = H(w)X(w)
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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