Discrete Fourier Transform, DFT Input: N time samples

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1 EE445M/EE38L.6 Lecture. Lecture objectives are to: The Discrete Fourier Transform Windowing Use DFT to design a FIR digital filter Discrete Fourier Transform, DFT Input: time samples {a n = {a,a,a 2,,a - Output: a set of frequency bins {A k = {A,A,A 2,,A - A k n a n W kn where Inverse DFT Input frequency bins {A k ={A,A,A2,,A - Output time domain samples {a n ={a,a,a 2,,a - a n k A k W -kn where W j2 / e k=,,2,,- W j2 / e n=,,2,,-

2 EE445M/EE38L.6 Lecture.2 While the DFT deals only with samples and bins, with no concept of seconds and Hz, when looking at ADC samples spaced at intervals T (in sec) Frequency bin m represents components at k*f S / (in Hz) The DFT resolution in Hz/bin is the reciprocal of the total time spent gathering time samples; i.e., /(T) for(n = ; n < 64; n++){ data = OS_Fifo_Get(); if(filterflag) data = FIR(data); if(voltflag) PlotData(data); x[n] = data&xffff; // real is to 23, imaginary part is cr4_fft_64_stm32(y,x,64); // y(k) has same units as x(n) for(k = ; k < 32; k++){ real = y[k]&xffff; // bottom 6 bits imag = y[k]>>6; // top 6 bits mag = sqrt(real*real+imag*imag); if(fftflag) LCD_Plot(mag); if V is the DFT output magnitude in volts db FS = 2 log (V/3); // full scale is 3. volts

3 EE445M/EE38L.6 Lecture.3 This is code from Lab2 consumer showing how to run the FFT for(t = ; t < 64; t++){ // collect 64 ADC samples data = OS_Fifo_Get(); // get from producer x[t] = data; // real to 23, imaginary cr4_fft_64_stm32(y,x,64);// complex FFT of ADC values If you want to calculate the magnitude from this FFT, the top 6 bits of y has the imaginary part and the bottom half is real. for(t = ; t < 32; t++){ // first half real = y[t]&xffff; // bottom 6 bits imag = y[t]>>6; // top 6 bits mag[t] = sqrt(real*real+imag*imag); This code takes 24 points, and plots 4 pixels per tick. This means there are 52/4=28 lines across the screen (like the cover of the book). It also uses the db full scale feature of the plotter for(t = ; t < 24; t++){ // collect 24 ADC samples data = OS_Fifo_Get(); // get from producer x[t] = data; // real to 23, imaginary cr4_fft_24_stm32(y,x,24); // complex FFT for(t = ; t < 52; t++){ // first half real = y[t]&xffff; // bottom 6 bits imag = y[t]>>6; // top 6 bits data = sqrt(real*real+imag*imag); ST7735_PlotdBfs(data); if((t%4)==3){ RIT28x96x4Plotext(); // 4 pixel per tick ST7735_Plotext(); //28 ticks across screen Applications Measure S/ ratio Identify noise DF design Four or Five approximations Finite min Finite max (range = max-min) Precision (resolution is range/precision) Sampling rate Finite number of samples -> Spectral leakage

4 EE445M/EE38L.6 Lecture.4 Inherent in the application of FFT or cross correlation in computer based systems is the need to operate on finite sequences. Even virtual memory has finite size, and most customers are not willing to wait for infinite time to get the results. The process of choosing a finite subsequence on which to operate is called ing. It is critical to capture a reasonable, because the data is actually considered as an infinite periodic signal. In other words, if we process the finite sequence x(), x(), x(2), x(-) then the FFT or cross correlation will effectively be determined for the infinite sequence, x(), x(), x(2), x(-), x(), x(), x(2), x(-), x(), x(), x(2), x(-), Figures 8.4a and 8.4b show an improperly chosen. otice that the infinite periodic signal does not accurately represent the original data. 2.5 EKG (mv) Time (seconds) Figure 8.4a. Original data with from 2 to 3 seconds 2.5 EKG (mv) Time (seconds) Figure 8.4b. Equivalent infinite periodic signal resulting from the shown in Figure 2a.

5 EE445M/EE38L.6 Lecture.5 Figures 8.5a and 8.5b show a properly chosen. otice that the infinite periodic signal accurately represents the shape of the original data, but the information about heart rate is lost. 2.5 EKG (mv) Time (seconds) Figure 8.5a. Original data with from.7 to 2.5 seconds.5 EKG (mv) Time (seconds) Figure 8.5b. Equivalent infinite periodic signal resulting from the shown in Figure 8.5a. If the has multiple cycles, then there will be multiple correlations. To prevent multiple matches, we can choose a with one cycle (like Figure 8.5), or we can a mask to the data, as shown in Figure 8.6. This is a trapezoidal mask, but other mask shapes can be used (rectangle, sine-wave, exponential). This method allows us to select a without having to specify the sequence length. otice that the shown in Figure 8.6 could be used to study the shape of just the QRS wave, without including the P and T waves.

6 EE445M/EE38L.6 Lecture.6.5 mask EKG (mv) Time (seconds) Figure 8.6. A is created by multiplying the original data with a mask.

7 EE445M/EE38L.6 Lecture.7 Windowing Spectral leakage can be virtually eliminated by ing time samples prior to the DFT Windows taper smoothly down to zero at the beginning and the end of the observation Time samples are multiplied by coefficients on a sample-by-sample basis Windowing sinewaves places the spectrum at the sinewave frequency Convolution in frequency

8 EE445M/EE38L.6 Lecture.8 - w(k) 2 = n= Window coefficients w(k) will be normalized so that the rms value of the time samples is the same before and after ing; that is, Hamming w(k) = *cos(2 k/(-)) Hann w(k) = (sin( k/(-))) 2 Cosine w(k) = sin( k/(-)) Triangle Reference 22 Eric Swanson, Mixed Signal Class w(k) = (2/)(/2 - k (-)/2 ) Window FFT Windowing Hamming Cosine Sample number FIR digital filter design You will process the real-time data in the foreground by implementing a FIR digital filter. After you have chosen the sampling rate (e.g., 44 khz) you next will choose a FIR filter length (e.g., =64). The ratio f s / (e.g., 44 khz/64 = 687 Hz) will determine the frequency resolution of the FIR filter design. Let H(z) be the desired filter

9 EE445M/EE38L.6 Lecture.9 gain transfer function. Table gives an example desired frequency response. The magnitude of H(k) is selected to implement the desired gain versus frequency response. In order to preserve the shape of the audio signals, we will implement linear phase. For frequencies above ½ fs, we make H(k) be the complex conjugate of the -k term. This will guarantee that the inverse DFT of H(z) will yield real results. The desired filter response, plotted as blue dots in Figure. 2 FIR digital filter 8 Gain Frequency Figure. Desired filter response. This is H. k f (Hz) Mag(H(k)) Angle(H(k)) k f (Hz) Mag(H(k)) Angle(H(k))

10 EE445M/EE38L.6 Lecture Table. Desired filter response for one patient that compensates for hearing loss. This is H. Let x(n) be the input (read from the ADC) and X(z) be the input in the frequency domain. Let y(n) be the FIR filter output, and let Y(z) be the FIR filter output in the frequency domain. Y(z) = H(z) X(z) y(n) = IFFT { H(z) FFT{x(t) Take Inverse FFT of the desired gain to get =64 FIR filter coefficients. Because the negative frequencies in Table are complex conjugates of the positive frequencies, h(n) will be real. h(n) = IFFT{H(z) =.93, ,.668, -4.3,.535, -3.53,.744, -2.,.744, -3.53,.535, -4.3,.668, ,.93, -.76,.9273, -.678, , -2.29, , -.78, , 2., , -.9, , , 5.826, , , , 34.28, , 2.636, , , , , 494., , , , , 2.636, , 34.28, , , , 5.826, , , -.9, , 2., , -.78, , -2.29, , -.678,.9273, -.76 Scale to make fixed point coefficients h to h63, e.g., /489 const long h[64]={489,-74,7,-27,37,-769,446,-52,446,-769,37,-27,7, -74,489,-3,237,-44,-667,-564,-58,-8,-767,52,-383,-256,-222, -7,49,-939,5655,-748,8755,-5827,5275,-777,-527,-322,-4434,26464,-4434, -322,-527,-777,5275,-5827,8755,-748,5655,-939,49,-7,-222,-256,-383, 52,-767,-8,-58,-564,-667,-44,237,-3; Multiplication in the frequency domain is equivalent to convolution in the time domain. The FIR filter is the convolution of the data with the inverse transform of the desired filter. y(n) = h(n) * x(n) = x(n) * h(n) ( * means convolution here) y(n) = sum [h(i) x(n-i)] as i goes from to +. ( means multiplication here) Because there are a finite number of h(n) terms, the convolution is a finite sum y[i]= (h[]*x[i]+h[]*x[i-]+h[2]*x[i-2]+ +h[63]*x[i-63])/256; // * means multiplication here

11 EE445M/EE38L.6 Lecture. What is the effect caused by sampling jitter?

Lab 4 Digital Scope and Spectrum Analyzer

Lab 4 Digital Scope and Spectrum Analyzer Page 4.1 Lab 4 Digital Scope and Spectrum Analyzer Goals Review Starter files Interface a microphone and record sounds, Design and implement an analog HPF, LPF