Problem 1: Digital Data Acquisition and Fourier Spectrum (40 points) A voltage signal is given by: f(t) = 2.0 + 2.5 sin(42.50πt) - 4.0 cos(18.50лt - 3л) V, where t is time in seconds. The signal is sampled by a DAQ with a 16-bit A/D converter. a) Calculate the range of sampling frequencies necessary to avoid aliasing, in Hz. b) Suppose the signal is sampled at 26 Hz for 3 seconds. Using Excel or MATLAB, plot f(t) as a smooth curve representing the analog signal from 0 to 3 seconds. On the same figure, plot the digitally acquired data as symbols connected by straight line segments. c) Predict the frequencies and corresponding amplitudes that would be seen on a frequency spectrum of these data. d) Sketch (by hand) the frequency spectrum that you would see. Pay careful attention to details such as the axes limits (e.g., folding frequency), the shape, amplitude, and frequency of spikes in the spectrum, etc. e) Perform an FFT to generate a frequency spectrum and list the frequencies and amplitudes of the peaks that you see. Attach your plots and any code you write. Does the FFT match your sketch from part (d)? Hint: You can generate your FFT using either Excel or MATLAB. For Excel, see the file "Excel_example_FFT_sine_wave.xls" and instructions in the learning module on FFTs. For MATLAB, see the file "MATLAB_example_FFT_sine_wave.m❞ f) Calculate the frequency resolution associated with the FFT. g) The A/D converter has the following range options: -10 to 10, -5 to 5, -2.5 to 2.5, -1 to 1, and 0 to 10 V. Which of these should be used to sample the signal and why? h) Calculate the resolution and quantization error for the 16-bit A/D converter and range option you selected in part (g), in mV.

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Problem 1: Digital Data Acquisition and Fourier Spectrum (40 points) A voltage signal is given by: f(t) = 2.0 + 2.5 sin(42.50πt) - 4.0 cos(18.50лt - 3л) V, where t is time in seconds. The signal is sampled by a DAQ with a 16-bit A/D converter. a) Calculate the range of sampling frequencies necessary to avoid aliasing,