Question

For a function f(x), its Fourier transform can be defined as g(y)=∫ −[infinity][infinity]​ f(x)e −i² πxydx. Use FFT, calculate (approximately) the Fourier transform of f(x)=[cos(6πx)+sin(7πx)]e −πx² . Suggestions: truncate x to [−L,L) for L=20 with N=2m=4096 points, truncate y to [−m/(2L),(m−1)/(2L)] with also 2m points, plot f(x) on [−2,2] and plot real and imaginary parts of g on [−7,7].

Answer 1

To calculate the Fourier transform of the given function using the Fast Fourier Transform (FFT) algorithm and plot the results.

Step-by-step explanation:

To calculate the Fourier transform of the given function using the Fast Fourier Transform (FFT) algorithm, we need to truncate the values of x and y to specific intervals. Let's set L = 20 and N = 4096 for the x-interval, and m = 2048 for the y-interval.

After truncating, we can plot the function f(x) on the interval [-2, 2] and the real and imaginary parts of the Fourier transform g(y) on the interval [-7, 7]. This will give us an approximate representation of the Fourier transform.