EE422G Signals and Systems Laboratory Fourier Series and the DFT Kevin D. Donohue Electrical and Computer Engineering University of Kentucky

Fourier Series “In 1822, Joseph Fourier, a French mathematician, discovered that sinusoidal waves can be used as simple building blocks to describe and approximate any periodic waveform including square waves. Fourier used it as an analytical tool in the study of waves and heat flow. It is frequently used in signal processing and the statistical analysis of time series.” Joseph Fourier

Sum of Sines and Square Wave Fourier Series analysis indicates that adding up odd harmonic sine waves with zero phase and weighted by 1/n (where n is the harmonic number) results in a square wave. Example for harmonics 1,3,5, and 7

Sum of Sines and Square Wave With 50 odd harmonics:

Dirichlet Conditions Given x(t) periodic with period T F, x(t) = x(t+nT F )  n  I and 1) (absolutely integrable). 2) x(t) has finite maxima and minima over a single period. 3) x(t) has a finite number of discontinuities all of finite size over T F.

Fourier Series Definition Then x(t) can be represented with a Fourier Series (FS): where (fundamental frequency) and complex Fourier series coefficient is:

Example Find the FS of a periodic train of impulse functions. Show result:

FS Numerical Computation Apply rectangular rule for numerical integration of: Consider N F samples taken at T S intervals over period T F where Then summing up the areas of rectangles over the integrand results in:

Discrete Fourier Transform (DFT) With minor changes, the numerical evaluation of the FS coefficients become the DFT defined as: Example: Find the DFT for the 4-point sequence x[] = { } Show

DFT Exercises For each discrete sequence below, compute the DFT by hand: x[] = {1, 1, 1, 1, 0, 0, 0, 0} x[] = {1, 0, -1, 0, 1, 0, -1, 0} x[] = {1, 0, -1, 0, 0, 0, 0, 0} Note that for N=8, the DFT kernel takes on only 8 possible complex number evenly spaced (by 45º) on the unit circle. Also, check answers with Matlab using: >> x = [ ] >> g = fft(x)

Inverse DFT (IDFT) The inverse DFT can be obtained directly from the FS expansion from it coefficients, where : Sample time axis and let which results in the IDFT: by convention the 1/N scale factor is removed from the DFT but used in the IDFT.

DFT Implications Points outside of the [0, N-1] interval for x[n] are assumed to be periodic since the FS was the starting point in the derivation. Therefore, the DFT coefficients represent a periodic extension of x[n] on [0, N-1]. x[n] = x[n+mN]  m  I Likewise the periodicity of the DFT kernel implies that the sampled time signal at f s = 1/T s on [0, T F ] results in a periodic extension of the spectrum or DFT coefficient on the interval [0, N-1] or [0 f s ) or [-f s /2, f s /2) (aliasing)  m  I

IDFT Exercises For each discrete sequence below, compute the DFT by hand: [] = {1, 1, 1, 1} [] = {1, -j, -1, j} [] = {1, 0, -j, 0, -1, 0, j, 0} Note that for N=8, the IDFT kernel takes on only 8 possible complex number evenly spaced (by 45º) on the unit circle. Also, check answers with Matlab using: >> g = [1, -j, -1, j] >> x = ifft(g)