- The Fourier transform is nothing but, it is a generalization of the Fourier series representation of functions.
- The advantage of Fourier transform over the Fourier series is that Fourier transform can be used for a larger class of functions which are not necessarily periodic
- Since the transform is essential to the understanding of several exercises,so here we will explain some basic Fourier transform concepts.
Recall the definition of the Fourier series expansion for a periodic function u(t). If u(t + T) = u(t), then
where ωm = 2πm/T so that
If u(t) is a function of time it is natural to call ωm an angular frequency and to refer to C(ωm) as the amplitude of u(t) at frequency ωm. Together, the c(ωm) comprise the frequency spectrum of u(t).
Computing the frequency spectrum for a sine or square wave of period T is fairly simple. For the sine wave, there are only two non-zero coefficients
- Since the odd fact that the formalism requires both positive and negative frequencies. The spectrum of the square wave is slightly more complicated
- The interesting feature here is that the rapid rise and fall of the square wave leads to spectral components at high frequency, as demonstrated in figure below.
- Building up a square wave from Fourier series components. Each graph is a plot of Sn, the sum of the first n terms in the series expansion, for increasing values of n.
- For many purposes a knowledge of the C(ωm) may be more useful than knowing u(t) itself. It is therefore desirable to extend the idea of a frequency spectrum to non-periodic functions, for example single pulses or finite wave trains. This can be done by replacing the sum of Eqn. 1 by an integral, giving
- where ω is now a continuous variable. The complex function c(ω) is called the Fourier transform of u(t) and is again the amplitude of u(t) at frequency ω.
- This can be understand as the transform of exp(iωot) is a delta function, δ(ω-ωo). This says that the spectrum of exp(iωot) has a large amplitude only at ω=ωo, which seems reasonable since the function is periodic with angular frequency ωo.
Example: Consider another specific, simple, and important example. Consider the “rect” function (“rect” for “rectangle”) defined by
The Fourier transform of a pulse centered at t = 0 ; is simply
- Note that a short pulse (small t) leads to a significant c(ω) over a wide frequency range, while a long pulse (large t) implies c(ω) decreases quickly with frequency.
- Π(t) is even — centered at the origin — and has width 1. Later we’ll consider shifted and scaled versions. Here You can think of Π(t) as modeling a switch that is on for one second and off for the rest of the time. Π is also called, variously, the top hat function (because of its graph), the indicator function, or the characteristic function for the interval (−1/2, 1/2).
- While we have defined Π(±1/2) = 0, other common conventions are either to have Π(±1/2) = 1 orΠ(±1/2) = 1/2.
- Some people don’t define Π at ±1/2 at all, leaving two holes in the domain. Here we don’t want to get dragged into this dispute. It almost never matters, though for some purposes the choice Π(±1/2) = 1/2 makes the most sense.
- In the above two plots of the Fourier coefficients of periodized rectangle functions with periods 2, 4, and 16, respectively. Because the function is real and even, in each case the Fourier coefficients are real, so these are plots of the actual coefficients, not their square magnitudes.
Since general function f(t) of period T the Fourier series has the form
Here the frequencies are 0, ±1/T, ±2/T, . . ..Points in the spectrum and are spaced 1/T apart,indeed in the pictures above the spectrum is getting more tightly packed as the period T increases.
- The n-th Fourier coefficient is given by
We can calculate this Fourier coefficient for Π(t):
- Here although the spectrum is indexed by n the points in the spectrum are n/T (n = 0, ±1, ±2,...), and it’s more easy to think about the “spectral information” (the value of cn) as a transform of Π evaluated at the points n/T.
- so we can write this, provisionally, as
- To compensate for this we scale up by T, that is, we consider instead
- Now, if T is large so we can replace the closely packed discrete points n/T by a continuous variable, say s, so that with s = n/T we would then write, approximately,
- This procedure look like in terms of the integral formula Simply as
- If we Consider as T → ∞ having the effect of replacing the discrete variable n/T by the continuous variable s, as well as pushing the limits of integration to ±∞, then we may write for the (limiting) transform of Π the integral expression such as
So the integral becomes
- So finally the graph is obtained now certainly see the continuous curve that the plots of the discrete, scaled Fourier coefficients are shadowing.
- The above function (sin πx)/πx comes up so often in this subject that it’s given a name, sinc.
- It pronounced as "Sink" ⇒ sinc0 = 1 ; Hence by virtue of the famous limit
- We now define the Fourier transform of a function f(t) as
Recovering f(t) from f(s)
- Again suppose f(t) is zero outside some interval and periodize it to have (large) period T. We expand f(t) in a Fourier series
- The Fourier coefficients can be written via the Fourier transform of f evaluated at the points sn = n/T
Here we can extend the limits to ±∞ since f(t) is zero outside of [−T/2,T/2]
So finally f(t) becomes
So that we have recovered f(t) from f(s).
The Inverse Fourier Transform
The integral we’ve just come up with can stand on its own as a “transform”, and so we define the inverse Fourier transform of a function 'g(s)' to be
Fourier Inversion Theorem
Again here we are treating this formally for a moment withholding a discussion of conditions under which the integral makes sense. In the same way we can produced the Fourier inversion theorem. That is
this can also be written as
- The Inverse Fourier Theorem looks just like the Fourier transform except for the minus sign.
Power Spectrum of Fourier integral
- The square magnitude |fˆ(s)|2 is called the power spectrum which is applied in communications the spectral power density or the energy spectrum.
- The relation between the energy of the signal in the time domain and the energy spectrum in the frequency domain is given by Parseval’s identity for Fourier transforms:
General Properties and Formulas
Convolution of Fourier Transform:
The convolution of two functions g(t) and f(t) is the function
We use the notation
Convolution Theorem: F(g∗f)(s) = Fg(s) Ff(s)