Fourier transform
fourier( returns the Fourier Transform
of f)f. By default, the function symvar
determines the independent variable, and w is the transformation
variable.
Compute the Fourier transform of common inputs. By default, the transform is
in terms of w.
| Function | Input and Output |
|---|---|
Rectangular pulse |
syms a b t f = rectangularPulse(a,b,t); f_FT = fourier(f) f_FT = - (sin(a*w) + cos(a*w)*1i)/w + (sin(b*w) + cos(b*w)*1i)/w |
Unit impulse (Dirac delta) |
f = dirac(t); f_FT = fourier(f) f_FT = 1 |
Absolute value |
f = a*abs(t); f_FT = fourier(f) f_FT = -(2*a)/w^2 |
Step (Heaviside) |
f = heaviside(t); f_FT = fourier(f) f_FT = pi*dirac(w) - 1i/w |
Constant |
f = a; f_FT = fourier(a) f_FT = pi*dirac(1, w)*2i |
| Cosine |
f = a*cos(b*t); f_FT = fourier(f) f_FT = pi*a*(dirac(b + w) + dirac(b - w)) |
| Sine |
f = a*sin(b*t); f_FT = fourier(f) f_FT = pi*a*(dirac(b + w) - dirac(b - w))*1i |
| Sign |
f = sign(t); f_FT = fourier(f) f_FT = -2i/w |
Triangle |
syms c f = triangularPulse(a,b,c,t); f_FT = fourier(f) f_FT = -(a*exp(-b*w*1i) - b*exp(-a*w*1i) - a*exp(-c*w*1i) + ... c*exp(-a*w*1i) + b*exp(-c*w*1i) - c*exp(-b*w*1i))/ ... (w^2*(a - b)*(b - c)) |
Right-sided exponential | Also calculate transform with condition f = exp(-t*abs(a))*heaviside(t); f_FT = fourier(f) assume(a > 0) f_FT_condition = fourier(f) assume(a,'clear') f_FT = 1/(abs(a) + w*1i) - (sign(abs(a))/2 - 1/2)*fourier(exp(-t*abs(a)),t,w) f_FT_condition = 1/(a + w*1i) |
Double-sided exponential | Assume assume(a > 0) f = exp(-a*t^2); f_FT = fourier(f) assume(a,'clear') f_FT = (pi^(1/2)*exp(-w^2/(4*a)))/a^(1/2) |
Gaussian | Assume assume([b c],'real') f = a*exp(-(t-b)^2/(2*c^2)); f_FT = fourier(f) f_FT_simplify = simplify(f_FT) assume([b c],'clear') f_FT =
(a*pi^(1/2)*exp(- (c^2*(w + (b*1i)/c^2)^2)/2 - b^2/(2*c^2)))/ ...
(1/(2*c^2))^(1/2)
f_FT_simplify =
2^(1/2)*a*pi^(1/2)*exp(-(w*(w*c^2 + b*2i))/2)*abs(c) |
Bessel of first kind with | Simplify the result. syms x f = besselj(1,x); f_FT = fourier(f); f_FT = simplify(f_FT) f_FT = (2*w*(heaviside(w - 1)*1i - heaviside(w + 1)*1i))/(1 - w^2)^(1/2) |
Compute the Fourier transform of exp(-t^2-x^2). By
default, symvar determines the independent variable, and
w is the transformation variable. Here,
symvar chooses x.
syms t x f = exp(-t^2-x^2); fourier(f)
ans = pi^(1/2)*exp(- t^2 - w^2/4)
Specify the transformation variable as y. If you specify
only one variable, that variable is the transformation variable.
symvar still determines the independent variable.
syms y fourier(f,y)
ans = pi^(1/2)*exp(- t^2 - y^2/4)
Specify both the independent and transformation variables as
t and y in the second and third
arguments, respectively.
fourier(f,t,y)
ans = pi^(1/2)*exp(- x^2 - y^2/4)
Compute the following Fourier transforms. The results are in terms of the Dirac and Heaviside functions.
syms t w fourier(t^3, t, w)
ans = -pi*dirac(3, w)*2i
syms t0 fourier(heaviside(t - t0),t,w)
ans = exp(-t0*w*1i)*(pi*dirac(w) - 1i/w)
Specify parameters of the Fourier transform.
Compute the Fourier transform of f using the default values
of the Fourier parameters c = 1, s = -1. For
details, see Fourier Transform.
syms t w f = t*exp(-t^2); fourier(f,t,w)
ans = -(w*pi^(1/2)*exp(-w^2/4)*1i)/2
Change the Fourier parameters to c = 1, s =
1 by using sympref, and compute the transform
again. The result changes.
sympref('FourierParameters',[1 1]);
fourier(f,t,w)ans = (w*pi^(1/2)*exp(-w^2/4)*1i)/2
Change the Fourier parameters to c = 1/(2*pi), s =
1. The result changes.
sympref('FourierParameters', [1/(2*sym(pi)), 1]);
fourier(f,t,w)ans = (w*exp(-w^2/4)*1i)/(4*pi^(1/2))
Preferences set by sympref persist through your current
and future MATLAB® sessions. Restore the default values of c and
s by setting FourierParameters to
'default'.
sympref('FourierParameters','default');Find the Fourier transform of the matrix M. Specify the
independent and transformation variables for each matrix entry by using matrices of
the same size. When the arguments are nonscalars, fourier acts
on them element-wise.
syms a b c d w x y z M = [exp(x) 1; sin(y) i*z]; vars = [w x; y z]; transVars = [a b; c d]; fourier(M,vars,transVars)
ans = [ 2*pi*exp(x)*dirac(a), 2*pi*dirac(b)] [ -pi*(dirac(c - 1) - dirac(c + 1))*1i, -2*pi*dirac(1, d)]
If fourier is called with both scalar and nonscalar
arguments, then it expands the scalars to match the nonscalars by using scalar
expansion. Nonscalar arguments must be the same size.
fourier(x,vars,transVars)
ans = [ 2*pi*x*dirac(a), pi*dirac(1, b)*2i] [ 2*pi*x*dirac(c), 2*pi*x*dirac(d)]
If fourier cannot transform the input then it returns an
unevaluated call.
syms f(t) w F = fourier(f,t,w)
F = fourier(f(t), t, w)
Return the original expression by using ifourier.
ifourier(F,w,t)
ans = f(t)
If any argument is an array, then fourier acts element-wise on all
elements of the array.
If the first argument contains a symbolic function, then the second argument must be a scalar.
To compute the inverse Fourier transform, use ifourier.
fourier does not transform piecewise.
Instead, try to rewrite piecewise by using the functions
heaviside, rectangularPulse, or
triangularPulse.
[1] Oberhettinger F., "Tables of Fourier Transforms and Fourier Transforms of Distributions." Springer, 1990.