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Computer Assignment 4.1

1. The following C program is to compute the magnitude and phase spectra of the signal

using the FFT function.

#include <stdio.h>

#include <math.h>#define N 256

void fft(double X[], double Y[], int npt, int inv);

void main()

{

int n;

double X[N], Y[N], magnit, phase; // X[N]:real part array, Y[N]:imaginary part

FILE *in, *out1, *out2; // in:input file, out1:magnitude, out2:phase

in = fopen("signal.dat", "r");

out1 = fopen("magnit.dat", "w");

out2 = fopen("phase.dat", "w");

for (n=0; n<N; n++)

{

fscanf (in, "%lf", &X[n]);

Y[n] = 0;

}

fft (X, Y, 256, 0); // X&Y: real and imaginary parts of signal and DFT

for (n=0; n<N; n++)

{

magnit = sqrt( X[n]*X[n] + Y[n]*Y[n] ); // Compute a magnitude response

phase = atan2(Y[n],X[n]); // Phase response

fprintf (out1, "%f\n", magnit);

fprintf (out2, "%f\n", phase);

}

fclose(in);

fclose(out1);

fclose(out2);

}

#define PI 3.14159265359

void fft( double xr[], double xi[], int npt, int inv )

{

int i, j, k, irem, m, l, le, le1, ip, sign;

double ur, ui, wr, wi, tr, ti, temp;

j = 0;

for (i=0; i<(npt-1); i++)

{

if (i<j)

{

tr = xr[j]; ti = xi[j];

xr[j] = xr[i];

xi[j] = xi[i];

xr[i] = tr; xi[i] = ti;

}

k = npt/2;

while (k<=j)

{

j = j-k;

k = k/2;

}j = j + k;

}

m = 0;

irem = npt;

while (irem>1)

{

irem = irem/2;

m = m + 1;

}

if (inv==1) sign = 1;

else sign = -1;



le = 1;

for (l=1; l<=m; l++)

{

le = le*2;

le1 = le/2;

ur = 1.0; ui = 0;

wr = cos(PI/le1);

wi = sign * sin(PI/le1);

for (j=0; j<le1; j++){

i = j;

while(i < npt)

{

ip = i + le1;

tr = xr[ip]*ur - xi[ip]*ui;

ti = xi[ip]*ur + xr[ip]*ui;

xr[ip] = xr[i] - tr;

xi[ip] = xi[i] - ti;

xr[i] = xr[i] + tr;

xi[i] = xi[i] + ti;

i = i + le;

}

temp = ur*wr - ui*wi;

ui = ui*wr + ur*wi;

ur = temp;

}}

if(inv == 1)

{

for (i=0; i<npt; i++)

{

xr[i] = xr[i]/npt;

xi[i] = xi[i]/npt;

}

}}

2. The following C program is to design a 16-point half-band lowpass filter using the

Blackman window and computes the frequency response of the filter.

#include <stdio.h>

#include <math.h>

#define N 1024

#define PI 3.14159265359

void fft(double xr[], double xi[], int npt, int inv);

void main()

{

int n;

double hrec[16], hbla[16], xr[N], xi[N], mag, pha, arg;

double norm;

FILE *filter, *out;

filter = fopen("filter.dat", "w");

out = fopen("blac.dat", "w");

for (n=0; n<=7; n++) // LPF design with rectangular window

{

arg = (.5 + n) * 0.5 * PI; // argument for sinc function

hrec[8+n] = .5 * sin(arg)/arg; // 0.5*sinc(arg)hrec[7-n] = hrec[8+n]; // Use even symmetric property

}

for (n=0; n<=15; n++) // Compute LPF coefficients using Blackman window

{

hbla[n] = hrec[n]*(.42 - .5*cos(2*PI*(n+1)/17) + .08*cos(4*PI*(n+1)/17) );

}

norm = 0.;

for (n=0; n<=15; n++) norm = norm + hbla[n]; // Find the frequency response at dc

for (n=0; n<=15; n++)

{

hbla[n] = hbla[n]/norm; // Normalize filter coefficients



fprintf (filter, "%d\t%f\n", n, hbla[n]);

}

for (n=16; n<N; n++) xr[n] = 0.0; // Pad extra zeros out

for (n=0; n<=15; n++) xr[n] = hbla[n]; // Copy filter coefficients

for (n=0; n<N; n++) xi[n] = 0.0; // Make imaginary part zero

fft (xr, xi, N, 0);

for (n=0; n<N; n++)

{

mag = xr[n]*xr[n] + xi[n]*xi[n];mag = 10*log10(mag);

pha = atan2(xi[n], xr[n]);

fprintf (out, "%f\t%f\n", mag, pha);

}

fclose(filter);

fclose(out);

}

The resulting filter coefficients are given below.

0 -0.000387

1 -0.002004

2 0.006093

3 0.014763

4 -0.031427

5 -0.062955

6 0.1321307 0.443787

8 0.443787

9 0.132130

10 -0.062955

11 -0.031427

12 0.014763

13 0.006093

14 -0.002004

15 -0.000387



0 50 100 150 200 250 300-4

-2

0

2

4

(a) signal

0 50 100 150 200 250 3000

50

100

150

200

(b) magnitude

0 50 100 150 200 250 300-4

-2

0

2

4

(c) phase

Fig. 3P-1 Signal and its magnitude and phase spectra

3. The magnitude and phase response of the filter are shown below.



0 200 400 600 800 1000 1200-150

-100

-50

0

lowpass magnitude response

0 200 400 600 800 1000 1200-40

-30

-20

-10

0

lowpass phase response

0 200 400 600 800 1000 1200-150

-100

-50

0

highpass magnitude response

Fig. 3P-2 Magnitude and phase response of the half-band lowpass filter and magnitude

response of the half-band highpass filter

4. The following C program is to implement the pair of lowpass and highpass filters.

#include <stdio.h>

#define N 16

#define N1 15

void main()

{

int n;

double buf[N], h[N], top, bottom, low, high;

FILE *in, *filt, *out1, *out2;

in = fopen("signal.dat", "r");

filt = fopen("filter.dat", "r");



out1 = fopen("lpf.dat", "w");

out2 = fopen("hpf.dat", "w");

for (n=0; n<N; n++) fscanf (filt, "%lf\n", &h[n]); // Read LPF coefficients

for (n=0; n<=N1; n++) buf[n] = 0.; // Initialize buffer with zero

while ( !feof(in) ) // As long as there is data in the file,

{ // the filtering process is continued.

fscanf (in, "%lf\n", &buf[0]);

top = bottom = 0.;

for (n=0; n<N; n=n+2) top = top + h[n]*buf[n]; // Even indexed coefficientsfor (n=1; n<N; n=n+2) bottom = bottom + h[n]*buf[n]; // Odd indexed coefficients

low = top + bottom; // Compute LPF output

high = top - bottom; // Compute HPF output

fprintf (out1, "%f\n", low);

fprintf (out2, "%f\n", high);

for (n=N1; n>0; n--) buf[n] = buf[n-1]; // Update buffer

}

fclose(in);

fclose(filt);

fclose(out1);

fclose(out2);

}

0 50 100 150 200 250 300-2

-1

0

1

2

lowpass output

0 50 100 150 200 250 300-2

-1

0

1

2

highpass output

Fig. 3P-3 Lowpass and highpass filter outputs

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