Georgia Tech, VISIBLE 3
Excerpt: ... Lab 19: The Fast Fourier Transform The goal of this lab is to introduce the Fast Fourier Transform (FFT) for efficient computer calculation of the Fourier Transform and then to investigate some of the Fourier Tranform's properties. ...
Colorado, ASEN 5519
Excerpt: ... . THE FAST FOURIER TRANSFORM 3 31 Chapter 3: THE FAST FOURIER TRANSFORM 32 3.1 FFT IN A NUTSHELL Even though this course is not about any numerical recipe, in order for us to utilize the numerical tools in intelligent ways, it pays to understand the essnetial features of the tools we would frequently use. the Fast Fourier Transform (FFT) is one such numerical tool. According to G. Strang, FFT is perhaps the most widely used numerical tool today in scientic and engineering computations. To start our discussion of FFT, let us recall MATLAB c version the Fourier transform: N 1 n=0 X (k) = x(s)e j2kn/N , k = 0, 1, . . . , N 1 (3.1) where the denominator 1/N is omitted for brevity. The one thing we have taken for granted is that for every kthe Fourier coefcient term, X (k), one must perform N-complex term multiplications and N additions, thus requiring N 2 multiplications in order to obtain the complete coefcients. Hence, for a measurement data set of 100 with 1,024 sample ...
Cornell, ECE 2200
Excerpt: ... The sampling frequencies proposed are: 16 kHz, 14 kHz, 12 kHz, 8 kHz, 5 kHz, 4 kHz. Problem 4 Consider the sequence x[n] = cos( 2 n) for n = 0, 1, ., 31. 16 (a) For what values of k is the DFT X[k] of the sequence x[n] non zero? Hint: Note that the other way of posing this problem is to consider the sequence x[n] as a vector x = (x[n])31 and the DFT transform as a matrix F32 . Then the DFT of x is n=0 X = F32 x. The question asks which components X[k] of the vector X are non zero. (b) Assume that in (a) we set all odd numbered frequency samples X[k] (odd components of X) to zero and call the new frequency sequence as Xodd [k] (or vector Xodd ). (The length of the vector Xodd remains 32). What is the corresponding to Xodd [k] the time sequence xodd [n]? (c) Repeat (b) when the even samples of X[k] are set to zero. 2 Problem 5 There exist digital circuits which implement Fast Fourier Transforms of a fixed size. Those are designed for ultimate speed that cannot be achieved in software. Assume that you ...