Mathematics 174 – Applied Mathematics 2 Discussion 5 – Properties of the Fourier Transform March 28, 2000

Background

We have introduced the Fourier Transform of a complex-valued f(x) on R:

F[f](ω) = \frac{1}{2π} ∫_{−∞}^{∞} f(x)e^{iωx} dx.

If f is piecewise continuous and in L^{1}(R) (that is, if ∫_{−∞}^{∞} |f(x)| dx < ∞), then the Fourier transform of f is another well-defined function on R.

For instance, we have seen that if

f(x) = \{ \begin{matrix}1 & if x ∈ [−a, a] \\ 0 & otherwise \end{matrix}

then

F[f](ω) = \frac{sin(aω)}{πω}.

The function \frac{sin(t)}{πt} is often called sinc(t), so the above formula for F[f](ω) could also be written as

F[f](ω) = asinc(aω).

Also, if f(x) = e^{−x^{2}} is the “standard” Gaussian, then

F[f](ω) = \frac{1}{\sqrt{4π}} e^{−ω^{2}/4}.

In addition, we have seen the general Shift Property of the Fourier Transform: If h(x) = f(x + a) is a horizontal shift of f, then

F[h](ω) = e^{−iaω}F[f](ω).

Comment: In communications and electrical engineering, the process of multiplying a function g(t) by an exponential e^{±iat} is called modulation. It corresponds to the idea of amplitude modulation (AM) for radio transmissions, where the signal g(t) would be multiplied onto a “carrier wave” cos(at) (the real part of e^{iat} = cos(at) + i sin(at)) before it is transmitted as g(t) cos(at).

Discussion Questions

A) Consider the function f(x) = e^{−|x|}. Show that f is piecewise continuous and in L^{1}(R), and compute its Fourier Transform.

B) Determine the following additional properties of the Fourier Transform and prove your statements.

1) What is the relationship between F[cf + g](ω), F[f](ω), and F[g](ω)?

2) What is the relationship between the Fourier Transform of f(x) and the Fourier transform of a modulation of f(x), h(x) = e^{iax}f(x)?

3) What is the relationship between the Fourier Transforms of f(x) and a horizontally scaled version of f(x), h(x) = f(cx)? Be careful about the sign of c!

C) If f is a real-valued function, then the real and imaginary parts of its Fourier Transform

are

\frac{1}{2π} ∫_{−∞}^{∞} f(x) cos(ωx) dx, \frac{1}{2π} ∫_{−∞}^{∞} f(x) sin(ωx) dx

respectively. Give a sufficient condition for the Fourier Transform of f to be real (imaginary part zero). Similarly, give a sufficient condition for the Fourier Transform of f to be purely imaginary (real part zero).

Assignment

Group write-ups due next class, Thursday, March 30.