We are often asked to confirm that a given filter design has the intended effect. This is often achieved using MATLAB or numpy $($or any similar computational environment$)$ using a white noise signalone that $($approximately$)$ contains all valid frequencies at approximately the same amplitude. In practice, we can generate this using a random number generator that produces random values between $0$ and $1-$in MATLAB this would be rand$-$shifting it to have a mean of zero $($i$.$e$.,$ subtract $0\mathrm{.}5$ from all values, such that the mean is $0$ and, therefore, $x[0]=0),$ passing the resultant zero$-$

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BME $3020$ C

Sensing and Measurement

Summer $2024$

mean signal through the convolutional kernel, and then evaluating how the frequency spectrum of the output differs from that of the input. We will here evaluate your filter from Problem $2$ in this manner.

a$.$ First, use rand $($or whatever is appropriate for your computational tooll$)$ to generate a sequence of $($at least$)2048$ random values having a zero mean. $($If you are using MATLAB, just subtract $0\mathrm{.}5$ from all values in the vector!$)$

b$.$ Use fft and fftshift to plot the magnitude of the frequency spectrum for your noise signal $($i$.$e$.,$ what you generated in part a$.).$ To make this a truly useful spectrum, you will want to use eftshift as well as to define an $x-$axis scale to use with plot. For this latter, recall that in an $N-$point DFT$,$ the $N^{th}$ value corresponds to $F_S($here taken to be $10$ kHz $),$ and the value at $\frac{N}{2}$ corresponds to one$-$half this value. Assuming you did not limit the length of the FFT performed, you should now have $($at least$)2048$ values that are to be mapped to encompass fin$[-\frac{F_S}{2},\frac{F_S}{2}).$ Note that this particular range assumes you have used fftshift. Once you do have a magnitude spectrum that covers fin$[-5000,5000)Hz,$ confirm the spectrum is "flat" in the sense that there is not a clear low$-,$ high$-,$ or band$-$pass$/$reject structure. $($If so$,$ re$-$run your random sequence generation process...you just got unlucky, which is improbable, but NOT impossible$).$

c$.$ Convolve your noise signal with the convolutional kernel you generated in Problem $2.$ If you are using MATLAB, be sure to use the 'same' option with conv to trim the result to match the length of the noise signal. $($Similar options exist in other computational environments!$)$

d$.$ Plot the magnitude of the frequency spectrum for your filtered signal. To make this a useful spectrum, you will once again want to use fftshift and define an x$-$axis. Once you have achieved an effective plot of the output frequency spectrum, compare the spectrum with that obtained in part b $.$ and confirm$/$refute that the passband corresponds to the targeted range of frequencies.