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ENTC 4337 Sine Generation

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Z-Transform Table Impulse n) 1 Unit Stepu(n) Rampnu(n) Exponentialanan Sinusoidal sin( nT) cos( nT)

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Digital Oscillator Design ? Impulse x(n)={1,0,0,...,0} Impulse Response h(n)=sin( nt)

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Design a digital filter that generates an 800 Hz sine wave in response to an impulse. Sampling frequency is 10,000 Hz. Test the filter by observing the first 10 output samples. Hint: Use the Z-transform table and convert H(z) to a difference equation.

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nx(n)x(n-1)y(n)y(n-1)y(n-2) 010000 1010.481800 2000.84430.48180 3000.99800.84430.4818 4000.90480.99800.8443 5000.58780.90480.9980 6000.12530.58780.9048 700-0.36810.12530.5878 800-0.7705-0.36810.1253 900-0.9823-0.7705-0.3681

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Types of Sequences Impulse function [x,n]=impseq(n0,n1,n2) %Generates x(n)=delta(n-n0); n1<=n<=n2 %------------------------------------- %[x,n]=impseq(n0,n1,n2) % n=[n1:n2]; x=[(n-n0)==0]; >>[x,n]=impseq(0,-3,9); >>stem(n,x); grid >>axis([0 10 0 2])

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Sinusoidal Sequence » n=[0:9]; » h=sin(0.503*n); » stem(n,h),grid

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Convolution » [x,n]=impseq(0,-3,9); » h=sin(0.503*n); » y=conv(x,h) y = Columns 1 through 7 0 0 0 -0.9981 -0.8447 -0.4821 0 Columns 8 through 14 0.4821 0.8447 0.9981 0.9042 0.5864 0.1233 -0.3704 Columns 15 through 21 -0.7723 -0.9829 0 0 0 0 0 Columns 22 through 25 0 0 0 0

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y MATLAB ={0,0.4821,0.8447,0.9981,0.9042,0.5864,0.1233,-0.3704,-0.7723,-0.9829} y HAND ={0,0.4821,0.8443,0.9980,0.9048,0.5878,0.1253,-0.3681,-0.7705,-0.9823}

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Design a digital filter that generates an 1000 Hz sine wave in response to an impulse. Sampling frequency is 10,000 Hz. Test the filter by observing the first 10 output samples. Hint: Use the Z-transform table and convert H(z) to a difference equation.

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>> [x,n]=impseq(0,-3,25); >> h=sin(0.628*n); >> y=conv(x,h); >> t=[-6:1:length(y)-7]; >> stem(t,y),grid MATLAB Implementation

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Note that 1 period is 10 samples and that 10 X.0001 seconds is 1 ms, or a frequency of 1 kHz.

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ax2 + bx + c 2x2 + 11x + 5 ac 2(5) = b =11 ( ) 2x2 + 10x + 1x

ax2 + bx + c 2x2 + 11x + 5 ac 2(5) = b =11 ( ) 2x2 + 10x + 1x

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