Fourier TransformΒΆ
Complex form or exponential form of Fourier Integral
\[
π\big[f(t)\big]=\int^{\infty}_{-\infty}{e^{-i\omega t}\ f(t)\ dt}
\]
assume that
\[
\begin{gather}
\int^{\infty}_{-\infty}{|f(x)|dx}
\end{gather}
\]
converges
-
Comparison
-
compare to Laplace transform
\[\begin{align} s \to i\omega\\\\ \int_0^{\infty} \to \int^{\infty}_{-\infty} \end{align}\] -
compare to Fourier Series Fourier Transform can be regarded as Fourier series whose \(p \to \infty\).
-
Amplitude SpectrumΒΆ
The amplitude spectrum of \(f(t)\) is a graph of \(\big|π\big[f(\omega)\big]\big|\)
Fourier & Inverse Fourier TransformΒΆ
-
Fourier transform
\[ π\big[f(x)\big]=\int^{\infty}_{-\infty}{e^{-i\omega x}\ f(x)\ dx} \] -
inverse Fourier transform
\[ π^{-1}\big[F(\omega)\big]=\frac{1}{2\pi}\int_{-\infty}^{\infty}{F(\omega)\ e^{i\omega x}\ d\omega}=f(x) \]
Fourier Sine TransformΒΆ
-
Fourier sine transform
\[ π_s\big[f(x)\big]=\int_0^{\infty}{f(x)\ \sin{(\omega x)\ dx=F(\omega)}} \] -
inverse Fourier sine transform
\[ π_s^{-1}\big[F(\omega)\big]=\frac{2}{\pi}\int_0^{\infty}{F(\omega)\,\sin{(\omega x)}\ d\omega}=f(x) \]
Fourier Cosine TransformΒΆ
- Fourier cosine transform
\[
π_c\big[f(x)\big]=\int_0^{\infty}{f(x)\,\cos{(\omega x)}\ dx}= F(\omega)
\]
- inverse Fourier cosine transform
\[
π_c^{-1}\big[F(\omega)\big]=\frac{2}{\pi}\int_0^{\infty}{F(\omega)\ \cos{(\omega x)}\ d\omega}=f(x)
\]
Fourier Transform propertiesΒΆ
LinearityΒΆ
\[
\begin{gather}
\alpha, \beta \in β \\\\
π\big[\alpha f(t)+\beta g(t)\big]=\alphaπ\big[f(t)\big]+π\big[\beta g(t)\big]
\end{gather}
\]
Time ShiftingΒΆ
\[
\begin{gather}
π\big[f(t-t_0)\big]=e^{-i\omega t_0} \ π\big[f(t)\big]
\end{gather}
\]
-
Inverse Time Shifting
\[ \begin{gather} π^{-1}\big[e^{-i\omega t_0}F(\omega)\big](t)=f(t-t_0) \end{gather} \]
Frequency ShiftingΒΆ
\[
\begin{gather}
π\big[e^{i\omega_0 t}\ f(t)\big]=F(\omega-\omega_0)
\end{gather}
\]
-
Inverse Frequency Shifting
\[ \begin{align} π^{-1}\big[F(\omega-\omega_0)\big]=e^{i\omega_0 t} \ f(t) \end{align} \]
ScalingΒΆ
\[
\begin{gather}
π\big[f(ct)\big]=\frac{1}{|c|}F(\frac{\omega}{c})
\end{gather}
\]
- Inverse Scaling
\[
\begin{gather}
π^{-1}\big[F(\frac{\omega}{c})\big]=|c|\ f(ct)
\end{gather}
\]
Time ReversalΒΆ
\[
\begin{gather}
π\big[f(-t)\big]=F(-\omega)\\\\
π^{-1}\big[F(-\omega)\big]=f(-t)
\end{gather}
\]
SymmetryΒΆ
\[
\begin{gather}
π\big[F(t)\big]=2\pi f(-\omega)
\end{gather}
\]
modulationΒΆ
\[
\begin{align}
π\big[f(t)\cos{(\omega_0 t)}\big]&=\frac{1}{2}[F(\omega+\omega_0)+F(\omega-\omega_0)]\\\\
π\big[f(t) \sin(\omega_0 t)\big]&=\frac{i}{2}[F(\omega+\omega_0)-F(\omega-\omega_0)]
\end{align}
\]
particular exampleΒΆ
prof. Laplace Transform
\[
\begin{gather}
π\big[H(t)e^{-at}\big]=\frac{1}{a+i\omega}
\end{gather}
\]
Frequency DifferentiationΒΆ
\[
\begin{gather}
π\big[t^nf(t)\big]=(i)^nF^{(n)}
\end{gather}
\]
Transforms of DerivativesΒΆ
-
for Fourier transform
\[ \begin{align} π\big[f'(x)\big]=i\omega F(\omega) \\\\ π\big[f^{(n)}\big]=(i\omega)^{n}\ F(\omega) \end{align} \] -
for Fourier sine transform
\[ \begin{align} π_s\big[f'(x)\big]=-\omega \ π_c\big[f(x)\big] \\\\ π_s\big[f''(x)\big]= -\omega^2\ π_s\big[f(x)\big]+\omega f(0) \end{align} \] -
for Fourier cosine transform
\[ \begin{align} π_c\big[f'(x)\big]=\omega \, π_s\big[f(x)\big]-f(0) \\\\ π_c\big[f''(x)\big]=-\omega^2π_c\big[f(x)\big]-f'(0) \end{align} \]
Transform of IntegralΒΆ
\[
\begin{align}
π\left[\int_{-\infty}^{t}{f(\tau )d\tau}\right]=\frac{1}{i\omega}F(\omega)
\end{align}
\]
ConvolutionΒΆ
- Definition
\[
\begin{gather}
f(t)*g(t)=\int_{-\infty}^{\infty}{f(\tau)\ g(t-\tau)\ d\tau}
\end{gather}
\]
- Property
\[
\begin{align}
f * g &= g * f \\\\
(\alpha f+\beta g)*h&=\alpha(f*h)+\beta(g*h)
\end{align}
\]
- Theorem
\[
\begin{gather}
\int_{-\infty}^{\infty}{\left[f(t)*g(t)\right]\ dt}=\int^{\infty}_{-\infty}{f(t)dt}\cdot \int^{\infty}_{-\infty}{g(t)dt}\\\\
π\big[f(t) * g(t)\big]=F(\omega) \ G(\omega)\\\\
π\left[f(t)\ g(t)\right]=\frac{1}{2\pi}F(\omega)*G(\omega)
\end{gather}
\]
Dirac Delta FunctionΒΆ
-
Definition
\[ \begin{gather} \delta(t)=\lim_{a\to 0}{\left[\frac{1}{2a}H(t+a)-H(t-a)\right]} \end{gather} \]
Fourier TransformΒΆ
\[
\begin{gather}
π\left[\delta(t)\right]=π\left[\lim_{a\to 0}{\left[\frac{1}{2a}H(t+a)-H(t-a)\right]}\right]=\lim_{a\to 0}{\frac{\sin{(a\omega)}}{a\omega}}=1
\end{gather}
\]
FilteringΒΆ
\[
\begin{gather}
\int^{\infty}_{-\infty}{f(t)\delta(t-t_0)\ dt}=f(t_0)
\end{gather}
\]
Discrete Fourier TransformΒΆ
Let \(u=\left\{u_j\right\}^{N-1}_{j=0}\)
\[
\begin{gather}
D[u]=U_k=
\sum_{j=0}^{N-1}{u_j\exp{\left(-i\ \omega_0 j\ \frac{T}{N}\right)}}
= \sum_{j=0}^{N-1}{u_j\exp{\left(-i\ 2\pi\ j\ \frac{1}{N}\right)}}
\end{gather}
\]
LinearityΒΆ
\[
\begin{gather}
D\left[au+bv\right]=aU_k+bV_k
\end{gather}
\]
PeriodicityΒΆ
\[
\begin{gather}
U_{k+N}=U_k
\end{gather}
\]
Inverse N-point DFTΒΆ
\[
\begin{gather}
u_j=\frac{1}{N}\sum_{j=0}^{N-1}{U_k\exp{\left(i\ \omega_0 j\ \frac{T}{N}\right)}}=\frac{1}{N}\sum_{j=0}^{N-1}{U_k\exp{\left(i\ 2\pi \ j\ \frac{1}{N}\right)}}
\end{gather}
\]
Complex Fourier CoefficientsΒΆ
\[
\begin{gather}
d_k \approx \frac{1}{N}\cdot U_k
\end{gather}
\]
note that the coefficient of complex Fourier \(C_\omega\)
\[
\begin{gather}
C_\omega=\frac{1}{2\pi}\cdot π\left[f(t)\right]
\end{gather}
\]
Sampled Fourier SeriesΒΆ
\[
\begin{gather}
S_M\left(\frac{jT}{n}\right)\approx\frac{1}{N}\sum_{k=0}
^{N-1}{V_k\ e^{i\omega_0 jkT/N }}
\end{gather}
\]
in which
\[
V_k=\left\{
\begin{align}
&U_k &\text{for }k=0, 1, \dots, M
\\\\
&0 &\text{for }k=M+1, \dots, N-M+1
\\\\
&U_k &\text{for }k=N-M, \dots, N-1
\end{align}\right.
\]
Solving the BVPΒΆ
- BVP (Boundary Value Problem)
There are three possible condition below
-
\(-\infty < x <\infty\)
\(\Rightarrow\) Fourier transform
-
\(0 < x < \infty\) and \(u(x,y)\bigg|_{x=0} = 0\)
\(\Rightarrow\) Fourier sine transform
-
\(0 < x < \infty\) and \(\frac{\partial}{\partial x}u(x, y)\bigg|_{x=0}=0\)
\(\Rightarrow\) Fourier cosine transform
Method of Separation of VariableΒΆ
kernel : assume that 1.
\[u(x,y)=X(x)\ Y(y)\]
2.
\["f(x)"\ =\ "f(y)"\ =\ -\lambda\]