Fourier transforms

Fourier analysis is a useful tool that only works on periodic signals. To transform non-periodic signals to periodic ones suitable for analysis, we use the Fourier transform, which is a subset of the more general Laplace transform.

Conceptually, the time-domain signal x(t)x(t) is the amplitude of the signal sampled at time instants infinitesimally close to each other, while the frequency-domain representation as the summation of individual contributions of all the frequencies, which are in-fact the amplitudes of the terms in the exponential Fourier series

The transform integral is given in the frequency domain as

X(ω)=F{x(t)}=lim⁡T0→∞[ck]T0=lim⁡T0→∞[∫−T02T02x(t)e−jkΔωt dt]=∫−∞∞x(t)e−jωt dt\begin{align*} X(\omega)=\mathcal{F}\{x(t)\}&=\lim_{T_{0}\to{\infty}}{[c_{k}]T_{0}}\\ &=\lim_{T_{0}\to{\infty}}{\left[\int_{-\frac{T_{0}}{2}}^{\frac{T_{0}}{2}}{x(t)e^{-jk\Delta\omega{t}}\,dt}\right]} \\ &=\int_{-\infty}^{\infty}{x(t)e^{-j\omega{t}}\,dt} \end{align*}

The inverse of this transform derives from the aforementioned exponential representation

x(t)=F−1{X(ω)}=lim⁡T0→∞[x~(t)]=12π∫−∞∞X(ω)ejωt dω\begin{align*} x(t)=\mathcal{F}^{-1}\{X(\omega)\}&=\lim_{T_{0}\to\infty}[\tilde{x}(t)]\\ &=\frac{1}{2\pi}\int_{-\infty}^{\infty}{X(\omega)e^{j\omega{t}}\,d\omega} \end{align*}

The existence of a Fourier transform for a given function (whether ϵ(t)\epsilon(t) converges 0 around non-discontinuities) is governed by the same Dirichlet conditions as the Fourier series.

Properties of FT

Linearity

The transformation is a linear operator, with the following relationship holding true

α1x1(t)+α2x2(t)+⋯+αnxn(t)↔α1X1(ω)+α2X2(ω)+⋯+αnXn(ω)\alpha_{1}x_{1}(t)+\alpha_{2}x_{2}(t)+\dots+\alpha_{n}x_{n}(t)\leftrightarrow\alpha_{1}X_{1}(\omega)+\alpha_{2}X_{2}(\omega)+\dots+{\alpha_{n}X_{n}(\omega)}

Duality

For the above definition of x(t)x(t) and X(ω)X(\omega) in terms of each other, if we swap the roles of functions, we get the following relationships

X(t)↔2πx(−ω)=x(−f)X(t)\leftrightarrow{2\pi{x(-\omega)}}=x(-f)

Symmetry

Transforms of odd and even signals

Similar to the Fourier series, for even and functions functions

ℑ(X(ω))=0       when x(t) is even ∀ω\Im(X(\omega))=0\;\;\;\text{ when }x(t)\text{ is even }\forall\omega ℜ(X(ω))=0       when x(t) is odd ∀ω\Re(X(\omega))=0\;\;\;\text{ when }x(t)\text{ is odd }\forall\omega

Time and frequency shifting

Time shifting is also similar to the Fourier series, given as

x(t−τ)↔X(ω)e−jωτx(t-\tau)\leftrightarrow{X(\omega)e^{-j\omega\tau}}

This also goes the other way, when shifting by frequencies

x(t)ejω0t↔X(ω−ω0)x(t)e^{j\omega_{0}t}\leftrightarrow{X(\omega-\omega_{0})}

Modulation

Multiplying the time function by cosine or sine causes it to be modulated, essentially its frequencies are shifted and scaled by 12\frac{1}{2}, in the case of cosine it shifts by its frequency, and sine does the same as well as an additional time shift by −π/2-\pi/2

x(t)cos⁡(ω0t)↔12[X(ω−ω0)+X(ω+ω0)]x(t)\cos(\omega_{0}t)\leftrightarrow{\frac{1}{2}}[X(\omega-\omega_{0})+X(\omega+\omega_{0})] x(t)sin⁡(ω0t)↔12[X(ω−ω0)e−jω2−X(ω+ω0)ejω2]x(t)\sin(\omega_{0}t)\leftrightarrow\frac{1}{2}[X(\omega-\omega_{0})e^{\frac{-j\omega}{2}}-X(\omega+\omega_{0})e^{\frac{j\omega}{2}}]

Time and frequency scaling

This property states that when one variable is scaled by a factor, the other variable is also scaled for the same factor, in the opposite direction, giving us

x(at)↔1∣a∣X(ωa)x(at)\leftrightarrow{\frac{1}{|a|}X(\frac{\omega}{a})}

Differentiation in time and frequency

Differentiating the time domain function to the nn-th degree gives us

dndtnx(t)↔(jω)nX(ω)\frac{d^{n}}{dt^{n}}x(t)\leftrightarrow(j\omega)^{n}X(\omega)

Similarly for the frequency domain

(−jt)nx(t)↔dndωn(X(f))(-jt)^{n}x(t)\leftrightarrow \frac{d^{n}}{d\omega^{n}}(X(f))

Convolution theorem (and multiplication)

To simplify solving convolutions, we can convert to the frequency domain

x1(t)∗x2(t)↔X1(ω)X2(ω)x_{1}(t)*x_{2}(t)\leftrightarrow X_{1}(\omega)X_{2}(\omega)

The opposite is also true

x1(t)x2(t)↔12πX1(ω)∗X2(ω)x_{1}(t)x_{2}(t)\leftrightarrow\frac{1}{2\pi}X_{1}(\omega)*X_{2}(\omega)

Integration

For a running integral of a time domain function

∫−∞tx(λ) dλ↔X(ω)jω+πX(0)δ(ω)\int_{-\infty}^{t}{x(\lambda)\,d\lambda}\leftrightarrow\frac{X(\omega)}{j\omega}+\pi{X(0)}\delta(\omega)

Energy and power

Parseval’s theorem

For periodic power signals with period T0T_{0} and EFS coefficient ckc_{k}, it can be stated

1T0∫t0t0+T0∣x~(t)∣2 dt=∑k=−∞∞∣ck∣2\frac{1}{T_{0}}\int_{t_{0}}^{t_{0}+T_{0}}{|\tilde{x}(t)|^{2}}\,dt=\sum\limits_{k=-\infty}^{\infty}{|c_{k}|^{2}}

and for a non-periodic energy signal, it can be stated

∫−∞∞∣x(t)∣2 dt=∫−∞∞∣X(f)∣2 df\int_{-\infty}^{\infty}{|x(t)|^{2}\,dt}=\int_{-\infty}^{\infty}{|X(f)|^{2}\,df}

Fast Fourier Transform