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Fundamentals of Signal Sampling and Rate Conversion

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Learn about sampling, Fourier transform, spectrum, reconstruction, downsampling, upsampling, filtering, ADC, DAC, and practice examples in signal processing.

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Fundamentals of Signal Sampling and Rate Conversion

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  1. Lab. 4 Sampling and Rate Conversion • Sampling: • The Fourier transform of an impulse train is still an impulse train. • Then, xs (t) x(nT) x(t) x * An impulse is an analog signal.

  2. Spectrum: • Reconstruction: Sampling xs (t) x(nT) x(nT) Ideal LPF x

  3. Practical reconstruction device (DAC):

  4. Practical sampling device (ADC): * FLASH ADC

  5. Ramp counter ADC:

  6. Successive approximation ADC: * Tree search

  7. Downsampling: • Let m=i+kM and we have Xd(n)= x(Mn) x(n) 

  8. Spectrum: • To avoid aliasing, a filter is generally applied before the downsampling operation. • Upsampling: i=1 i=1 i=0 i=0 i=0 Xd(n)= x(Mn) x(n) LPF Cutoff=/M  Gain=1 Xu(n)= x(n/L) x(n) 

  9. The spectrum: Ideal LPF

  10. The upsampling process is then equivalent to increase the sampling rate by a factor of L. • The filtering operation is also known as interpolation. Gain=L Xu(n) x(n) LPF Cutoff=/L 

  11. Practice 1: • Generate a sinusoidal signal, downsample the signal, and observe the its spectrum. • Determine the maximum downsampling rate such that the aliasing will not occur. • Then upsample the downsampled signal, and observe its spectrum.

  12. General filter design: • Pass band • Stop band • Transition band • Passband ripple/stopband ripple A lowpass filter

  13. The analog filter design (IIR): • 1. Butterworth, 2. Chebychev I, 3. Chebychev II, 4. Ellipic

  14. Fdatool in Matlab:

  15. Practice 2: • Generate a sinusoidal signal, downsample the signal (no aliasing), and then upsample the downsampled signal. • Design an FIR LPF and let the upsampled signal pass the filter such that the upsampled signal is similar to the original signal. • Calculate the MSE of these two interpolation schemes.

  16. Practice 3: • Create an random digital signal and upsampled it with a selected factor. • Observe the spectrum of the upsampled signal. • Reading assigment: • Pulse shaping (CS: 4.5) • RC, SRRC

  17. Filter design(FDA tool) • Key “fdatool” in the console of MATLAB • Adjust parameters for your requirement • Press “Filter coefficients” to get filter time-domain response h[n] • Convolve h[n] in your C program to implement lowpass filtering • Plot spectrum in MATLAB • Plot( abs( fft( x ) ) ) • fft(): Fast Fourier Transform, frequency interval is [0 fs] • abs(): get magnitude

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