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Digital Filters

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A/DComputerD/A x(t)x[n]y[n]y(t) Example:

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To get the current output y[n], which is sent to the D/A, we get the previous output y[n-1] (sent the last time to the D/A) and add to it the current input from the A/D. In a programming language like C++, the code would look something like this: yp = 0;// set “previous” y to zero while (true) { x = inp(ad);// get new sample y = yp+x;// perform filtering operation outp(y, da);// send filtered sample to D/A yp = y;// set “current” to “previous” // for next iteration }

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Finding the z-transform transfer function:

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Now what does this operation do to an input signal? Example: find the impulse and step responses to the previous filter. Solution: we could use z-transforms or convolution to find the output when the input is an impulse or a step, but instead, let us see how the output “evolves.” A table will be created with three columns: n, x[n] and y[n]. Each column will correspond to an instance in discrete-time. For the impulse input x[n]= [n] we have

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n x[n]= [n] y[n] 011 101 201 301 401

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For the step input x[n]=u[n] we have nx[n]=u[n]y[n] 011 112 213 314 415

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The plots of these outputs are as follows: 1234 n y[n] for x[n]= [n]

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1234 n y[n] for x[n]= [n] 2 4 3 1 5

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Example: Find the impulse response and the step response for the following transfer function: Solution: From the definition of the transfer function we have

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Cross-multiplying and converting back to discrete- time domain we have In other words, the current output is equal to one- half times the previous output plus the current input.

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n x[n]= [n] y[n] 011 10 20 30 40

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For the step input x[n]=u[n] we have nx[n]=u[n]y[n] 011 11 21 31 41

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The plots of these outputs are as follows: 1234 n y[n] for x[n]= [n]

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1234 n y[n] for x[n]= [n] 1 2

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The step and impulse functions for this transfer function look like those for an RC low-pass filter. Suppose that we wished to construct a digital filter corresponding to an RC low-pass filter? Can we convert an analog filter to a digital filter?

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The Matched z-Transform In the matched z-transform digital filter design method we try to “match” the impulse response of the analog filter with that of the digital filter being designed. To match the impulse responses, we take the inverse Laplace transform of the analog filter H(s) h(t), then sample the impulse response h(t) h[n], then take the z-transform of the sampled impulse response to get the z-transform transfer function h[n] H(z).

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L -1 Z Once we have our z-transform transfer function H(z), we apply the definition of the transfer function to write our digital filter equations: sample Analog PrototypeDigital Filter

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Example: Use the matched filter design method to design the digital equivalent of an integrator. Solution: The analog transfer function is The inverse Laplace transform is

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We then sample the impulse response to get h[n]: Finally, we take the z-transform of the impulse response to get the digital filter transfer function.

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Finally, we apply the definition of the z-transform transfer function to get the relationship between the input of the digital filter x[n] and the output of the digital filter y[n].

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Applying this formula to an arbitrary input x[n], we have nx[n]y[n] 0x[0] 1x[1]x[0]+x[1] 2x[2]x[0]+x[1]+x[2] 3x[3]x[0]+x[1]+x[2]+x[3]

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We see that the output is the summation of the input. Thus the digital filter accurately represents the analog filter. The digital filter does not always accurately represent the analog filter as will be seen in the next example.

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Example: Use the matched filter design method to design the following transfer function: where c = s /4, and s is the sampling frequency. Solution: First, we find the impulse response

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Then we sample the impulse response: Then we take the z-transform of the (discrete-time) impulse response:

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