Fourier and Laplace Transforms

A.1 Fourier Transform

Let f(x) be an absolutely integratable function; i. e., f—°TO F(x)dx < to, then F(x) and its Fourier transform F(a) are related by

TO to

F(a) = j F(x)e—laxdx, F(x) = j F(a)eiaxda.

— TO —TO

Derivative Theorem

By means of integration by parts, it is easy to find

TO TO

d F 1 d F la

— = — —e—laxdx = — F (x)e—laxdx = laF (a).

dx 2n dx 2n

— TO —TO

Shifting Theorem

TO

F(x + X)e—laxdx = 2^ j F(n)e—lan+laXdn = elaXF(a).

— TO

A.2 Laplace Transform

Let f(t) satisfy the boundedness condition /TO e—ct f (t)dt < to for some c > 0, then f(t) and its Laplace transform f(^) are related by

TO

f(oi) = 2П J f (t)elmtdt, f (t) = 2П J f (M)e—lmtdrn.

0 г

The inverse contour г is to be placed on the upper-half «-plane above all poles and singularities of the integrand (see Figure A1). This condition is necessary in order to satisfy causality.

Figure Al. Complex «-plane showing the inverse contour Г, • poles, ~~~~ branch cut.

Derivative Theorem

By applying integration by parts to the integral below, it is straightforward to show

Shifting Theorem

Let the initial conditions be

Ф, 0 < t < X 0, t < 0

then,

о о f(t + X _ 21 f f (t + X)eilotdt _ 2^ j f (nV^-^dn 0 X Г X 1 e~i«X f(v) – 2nj f (n)eiandn _ ^— f(v) + (1 – eicoX) 2п Ш 0

Similarly,

f(t – X _ eiaX f(co).