FACULTY OF ENGINEERINGMULTIMEDIA UNIVERSITY

ASSIGNMENT

ETM4096

DIGITAL SIGNAL PROCESSING

TRIMESTER 3 (2007/2008)

QUESTION 1

Part (a)(i)

Triangular window

Syntax:

triang(L)

Description:

triang(L) returns an L-point triangular window in the column vector w. The coefficients of a

triangular window are:

For L odd:

For L even:

The triangular window is very similar to a Bartlett window. The Bartlett window always ends

with zeros at samples 1 and L, while the triangular window is nonzero at those points. For L odd,

the center L-2 points of triang(L-2) are equivalent to bartlett(L)

Tukey window

Syntax :

w = tukeywin(L,r)

Description:

w = tukeywin(L,r) returns an L-point, Tukey window in column vector w. Tukey

windows are cosine-tapered windows. r is the ratio of taper to constant sections and is between 0

and 1. is a rectwin window and is a hann window. The default value for r is 0.5.

The equation for computing the coefficients of a Tukey window is

The window length is .

Gaussian window

Syntax :

w = gausswin(L)w = gausswin(L,α)

Description:

w = gausswin(L) returns an L-point Gaussian window in the column vector w. L is a

positive integer. The coefficients of a Gaussian window are computed from the following

equation.

where , and the window length is

w = gausswin(L,α) returns an L-point Gaussian window where α is the reciprocal of the

standard deviation. The width of the window is inversely related to the value of α; a larger value

of α produces a more narrow window. If α is omitted, it defaults to 2.5.

Chebyshev window

Syntax:

w = chebwin(L,r)

Description:

w = chebwin(L,r) returns the column vector w containing the length L Chebyshev window

whose Fourier transform sidelobe magnitude is r dB below the mainlobe magnitude. The default

value for r is 100.0 dB.

Bartlett window

Syntax:

w = bartlett(L)

Description:

w = bartlett(L) returns an L-point Bartlett window in the column vector w, where L must

be a positive integer. The coefficients of a Bartlett window are computed as follows:

For L odd

For L even

The window length .

The Bartlett window is very similar to a triangular window as returned by the triang function.

The Bartlett window always ends with zeros at samples 1 and n, however, while the triangular

window is nonzero at those points. For L odd, the center L-2 points of bartlett(L) are

equivalent to triang(L-2).

Bohman window

Syntax:

w = bohmanwin(L)

Description:

w = bohmanwin(L) returns an L-point Bohman window in column vector w. A Bohman

window is the convolution of two half-duration cosine lobes. In the time domain, it is the product

of a triangular window and a single cycle of a cosine with a term added to set the first derivative

to zero at the boundary. Bohman windows fall off as 1/w4.

The equation for computing the coefficients of a Bohman window is

where and the window length is .

Parzen window

Syntax:

w = parzenwin(L)

Description:

w = parzenwin(L) returns the L-point Parzen (de la Valle-Poussin) window in column

vector w. Parzen windows are piecewise cubic approximations of Gaussian windows. Parzen

window sidelobes fall off as .

Blackman-Harris window

Syntax:

w = blackmanharris(L)

Description:

w = blackmanharris(L) returns an L-point, minimum , 4-term Blackman-Harris window

in the column vector w. The window is minimum in the sense that its maximum sidelobes are

minimized.

The equation for computing the coefficients of a minimum 4-term Blackman-harris window is

where and the window length is .

The coefficients for this window are

a0 = 0.35875

a1 = 0.48829

a2 = 0.14128

a3 = 0.01168

Flat top windows

Syntax:

w = flattopwin(L)

w = flattopwin(L,sflag)

Description:

Flat Top windows have very low passband ripple (< 0.01 dB) and are used primarily for

calibration purposes. Their bandwidth is approximately 2.5 times wider than a Hann window.

w = flattopwin(L) returns the L-point symmetric flat top window in column vector w.

w = flattopwin(L,sflag) returns the L-point symmetric flat top window using sflag

window sampling, where sflag is either 'symmetric' or 'periodic'. The

'periodic' flag is useful for DFT/FFT purposes, such as in spectral analysis. The DFT/FFT

contains an implicit periodic extension and the periodic flag enables a signal windowed with a

periodic window to have perfect periodic extension. When 'periodic' is specified,

flattopwin computes a length L+1 window and returns the first L points. When using

windows for filter design, the 'symmetric' flag should be used.

Flat top windows are summations of cosines. The coefficients of a flat top window are computed

from the following equation

where and elsewhere and the window length is L = N +1. The coefficient values

are

a0 = 0.21557895 a1 = 0.41663158 a2 = 0.277263158

a3 = 0.083578947 a4 = 0.006947368

Part (a)(ii)

The difference between the Bartlett and Triangular window is that the Bartlett window always

ends with zeros at samples 1 and n, however, the triangular window is non-zero at those points.

Part (a)(iii)

The sinusoidal windows besides Hanning, Hamming and Blackman windows are:

- Bohman window

- Parzen window

- Flat top window

- Blackman-Harris window

- Bartlett-Hann

- Nutall Blackman Harris window

- Gaussian window

- Kaiser

- Lanczos

- Chebyshev

- Cosine

Part (a)(iv)

Similarities:

Rectangular, Tukey, Gaussian, and Kaiser yield fast roll-off in the frequency domain, but have

limited attenuation in the stop-band along with poor group delay characteristics. We can also see

that these windows have a narrow main lobe and significant ammount of side lobes.

Relation with rectangular window:

If we keep on increasing the width of these windows, the window pattern approaches rectangular window.

Part (b)(i)

Given = = 0.01, = 0.34 , = 0.36

2 f = -

2 f = (0.36 – 0.34)

f = 0.01

= - 20log

= -20log(0.01)

= 40dB

N =

N =

N = 224.

Since is in the range 21 , is defined as;

= 0.5842 + 0.07886( – 21)

= 0.5842 + 0.07886(40 – 21)

= 3.4

To plot the resulting window using the value obtained previously for and N, we write the

MATLAB code as below:

N = 224;

beta = 3.4;

w = Kaiser(N, beta)

plot(w)

Figure 1: Kaiser Window for = 3.4 and N = 224

Part (b)(ii)

MATLAB codes:

N = 224;

beta = 3.4;

n = 0:N-1

bes_1 = abs(besseli(0, beta));

x = beta*sqrt(1-[(n-round(N/2))/(round(N/2))].^2)

bes_2 = besseli(0,x);

w = bes_2./bes_1

length(w)

plot(w,'-r')

xlabel('samples');

ylabel('amplitude');

Figure 2: Kaiser Window obtained from the zeroth-order modified Bessel function of the first kind

Answer for this part agrees with the answer for the previous part

Part (b) (iii)

The desired filter is obtained by using the following code

N = 224;

beta = 3.4;

wc = 0.35;

h = fir1(N, wc,'high', kaiser(N+1, beta))

To plot the impulse response and the frequency response we write the codes

impz(h)

freqz(h)

Figure 3: Impulse response of the filter

Figure 4: Frequency response of the filter

Part ( c)

MATLAB code for designing the filter using Parks-McClellan algorithm:

Rp = 1;

Rs = 40;

fs = 8000;

f = [1500 2000];

a = [1 0];

dev = [(10^(Rp/20)-1)/(10^(Rp/20)+1) 10^(-Rs/20)];

[N,fo,mo,w] = remezord(f, a, dev, fs)

b = remez(N, fo, mo, w);

impz(b);

figure;

[h,f] = freqz(b, 1, 1024, fs);

freqz(b, 1, 1024, fs);

figure;

plot(f, (abs(h))) ;

Figure 5: Impulse response of the filter

Figure 6: Frequency response of the filter

Figure 7: Actual plot of frequency response

QUESTION 2

= 200kHz

First we calculate passband ripple, and stopband ripple,

1 = |20 (1 - )| 22 = |20 ( )|

1 - = =

In order to determine the order of the filter we need calculate discrimination factor, d and selectivity factor, k

d = 0.0405

= = 0.66667

The order, N is given as;

= = 7.909 = 8

The 3dB cutoff frequency, is calculated using

43.53

We set = 43.73kHz = 274776.9794rad/sec

Transforming using impulse invariant method:

MATLAT codes:

N = 8;

wc = 274776.9794;

fs = 200000;

[z, p, k] = buttap(N)

[num, den] = zp2tf(z, p, k)

[numt, dent] = lp2lp(num, den, wc)

[bz, az] = impinvar(numt, dent, fs)

Freqz(bz, az)

Figure 8: Filter obtained by using impulse invariant method

Transforming using bilinear transformation method:

MATLAB code:

N = 8;

wc = 274776.9794;

fs = 200000;

[z, p, k]= buttap(N)

[b, a] = zp2tf(z, p, k)

[num, den] = lp2lp(b, a, wc)

[bz, az] = bilinear(num, den, fs)

freqz(bz, az)

Figure 9: Filter obtained using bilinear transformation method