In systems, it is often necessary to analyze data across a certain domain. In the analog world, this domain is represented in the voltage versus time. In the digital world it may often become necessary to measure the signal’s spectral components, meaning the amplitude versus its frequency. The applications of such spectral measurements can be seen in testing communications system’s modulators to determine proper functioning; data and voice, and video compression for faster transmission. In particular, MPEG is a video compression algorithm using discrete cosine transforms to compress images and send them enabling real-time quality. You can also see its applications in radar signal processing, and pattern recognition. The science of the Discrete Fourier Transform stems from the publications of Fourier Analysis published in 1822. Fourier classifies signals as a sum of sine waves. The sine wave is considered the most basic signal of which other shaped signals can be developed. A square wave consists of one sine wave of one amplitude and frequency, and many other sine waves of higher frequencies and smaller amplitudes. The smaller sine waves are called Harmonics which are smaller representations of the sine wave but at different amplitudes. These signals are added together to produce the pulsed wave. To prove that the sine wave is the most basic of signals, construct an LC circuit and tune it to a particular frequency. Attach a signal generator outputting a square wave, triangle wave, or any other type of wave to the input of the LC circuit. Measure the output and you will see a sine wave at the frequency of the tuned LC circuit. The equation of the Fourier Series is represented below. f(t) = A0 + {[summation]|n=1 to infinity} An cos (nwt) + {[summation]|n=1 to infinity} Bn sin (nwt) where: A0 = 1/Tp {[integral]-Tp/2 to Tp/2} f(t) dt An = 2/Tp {[integral]-Tp/2 to Tp/2 f(t) cos (nwt) dt and Bn = 2/Tp {[integral]-Tp/2 to Tp/2 f(t) sin (nwt) dt A0 = amplitude of the main signal An = amplitude of the odd harmonic Bn = amplitude of the even harmonic nw = nth. Harmonic frequencies The series can be manipulated mathematically to become: f(t) = {[summation]n=-infinity to infinity} = Dn{[exp]-jwt} where: Dn = 1/Tp {[integral]-Tp/2 to Tp/2} f(t){[exp]-jnwt}dt The Fourier Transform for analog data is defined as: F(jw) = {[integral]-infinity to infinity} f(t){[exp]-jwt} The digital world uses discrete data rather than continuous data. To represent data in discrete form, a sample and hold circuit is used to capture continuous waveforms at specific time periods, known as sampling periods. These voltage impulses are then quantized and encoded into bit representations of 1’s and 0’s. This is now a digital representation of the signal where the discrete fourier transform can now analyze the frequency aspects of the discrete data. The formula for DFT is listed below. X(k) = Fd[x(nT)] = {[summation]n=0 to N-1}x(nT)[exp]-jk(Omega)nT = {[summation]n=0 to N-1}x(nT)[exp]-jk2[pi]n/N (simplified form) where: k=0,1,…N-1 The magnitude and phase angle is calculated as: |X(k)| = SQRT([R**2(k) + I**2(k)]) PHI(k) = [inverse]tan [I(k)/R(k)] (Omega) = 2[pi]/NT N = number of samples T sampling period Sometimes it is necessary to convert from the frequency domain to discrete time domain. To do this we implement the Inverse DFT which is listed below: X(k) = Fd(inverse)[x(nT)] = 1/N{[summation]n=0 to N-1}x(k)[exp] jk(Omega)nT where: n=0,1,…..,N-1 Here is an example of the DFT Using the sampled interval [1001] we will compute the Discrete fourier transform and its phase angle. X(0) = {[summation] n=0 to 3}x(nT) [exp] -j0 = {[summation] n=0 to 3}x(nT) = x(0) + x(T) + x(2T) + x(3T) = 1+0+0+1 = 2 X(1) = {[summation] n=0 to 3}x(nT) [exp] -j2[pi]n/N = 1+0+0+1[exp]-2[pi]3/4 = 1+[exp]-j3[pi]/2 = 1+cos (3[pi]/2) - jsin(3[pi]/2) = 1+j X(1) is complex with a magnitude of SQRT(2) and phase angle of (inverse)tan(1) = 45 degrees X(2) = {[summation] n=0 to 3}x(nT) [exp]-2n2[pi]/N = 1+0+0+1[exp]-j4[pi]3/4 = 1-1 = 0 X(3) = {[summation] n=0 to 3}x(nT) [exp] -j3n[pi]/N = 1+0+0+[exp]-j9[pi]/2 = 1-j X(3) is complex with a magnitude of SQRT(2) and phase angle of (inverse)tan(-1) = -45 degrees There fore, the time series is {1,0,0,1} and the DFT frequency series is: {2,1+j,0,1-j}. To plot the magnitude versus the angular frequency, (Omega) we first find Omega. To do this we must know the sampling rate and N. N = 4 in this case. Assuming a voice signal and sampled at 8KHz = 1/T, we have the frequencies where the magnitudes are placed. (Omega) = 12.57 kHz 2(Omega) = 25.14 kHz 3(Omega) = 37.71 kHz 4(Omega) = 50.28 kHz We also use these frequencies to plot the phase versus frequency as well.

- Title: Digital Signal Processing A Practical Approach