Sampling Theorem

A function f(t) in (L^2)(ℝ) is bandlimited if the support of its Fourier transform lies in [-Ω,Ω].  That is, the Fourier transform (OverHat[f])(ξ) equals zero for frequencies \[LeftBracketingBar]ξ\[RightBracketingBar]>Ω.

A fundamental result in Fourier analysis is the amazing Shannon sampling theorem:  If f(t) is bandlimited such that (OverHat[f])(ξ)=0 for all \[LeftBracketingBar]ξ\[RightBracketingBar]>Ω, then 

f(t) = (∑_(k=-∞)^∞)f((k/(2Ω)))sinc(2Ωt-k), 

with both pointwise and (L^2) convergence on ℝ.  This is wonderful because it tells us exactly how often to sample (every 1/(2Ω) in time) in order to have enough information to reconstruct the entire signal.

*Note that the textbook defines sinc(x)=sin(Pi x)/(Pi x) but Mathematica defines it to be sinc(x)=sin(x)/x, so be aware of the scaling factor difference.*

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Save this file with your work in the exercises below and email to tleise@amherst.edu by next Monday.

Exercise 1. Find the function f(t) whose Fourier transform is 1 for frequencies in [-2,2] and zero otherwise. Hint: use the InverseFourierTransform and UnitBox functions. What is the value of Ω of this function?, after Fall Break))

1.  Let f(x) equal 2x on [0,1/2] and 2-2x on (1/2,1].  (Use the Piecewise command.)
(a) Find general expressions for the Fourier coefficients using sines and cosines.  
(b) Plot the function and a partial sum on the same graph to check that your coefficients are correct.
(c) Find the sup norm of the difference between f and the partial sum of its Fourier series for n up to 5, then 10, then 20.  You may estimate the maximum error from a plot of Abs[f[x]-FS[x]].  If part of the graph is not shown, use Plot[fun,{x,a,b},PlotRange->All], and adjust the plot interval to focus on the spots with highest error.
(d) How many terms are needed in the partial sum so that the sup norm of the difference is less than 0.01?

Exercise 2. Find the function f(t) whose Fourier transform is 1 for frequencies in [0,4] and zero otherwise. What is the value of Ω of this function? How is it related to the function f(t) found in Exercise 1? (Hint: look at your Fourier transform table properties.), after Fall Break))

1.  Let f(x) equal 2x on [0,1/2] and 2-2x on (1/2,1].  (Use the Piecewise command.)
(a) Find general expressions for the Fourier coefficients using sines and cosines.  
(b) Plot the function and a partial sum on the same graph to check that your coefficients are correct.
(c) Find the sup norm of the difference between f and the partial sum of its Fourier series for n up to 5, then 10, then 20.  You may estimate the maximum error from a plot of Abs[f[x]-FS[x]].  If part of the graph is not shown, use Plot[fun,{x,a,b},PlotRange->All], and adjust the plot interval to focus on the spots with highest error.
(d) How many terms are needed in the partial sum so that the sup norm of the difference is less than 0.01?

Exercise 3. Recalling the connection between the smoothness of a function and the rate of decay of its transform, speculate how smooth a bandlimited function will typically be.

Exercise 4. Take the function f[t_]:=Sinc[6 Pi t]-Sinc[4 Pi t].  Let's try out the sampling idea.  First you need to determine the value of Omega for this signal. Next find partial sums of the series in the Shannon sampling theorem.  Note that f(0)=0, and sum separately over both positive and negative indices.  How many samples are needed to get a decent reconstruction of the original function, say a maximum pointwise error of 0.01 on the interval [-5,5] (you can plot the error to estimate this)?

f[t_]:=Sinc[6 Pi t]-Sinc[4 Pi t]
f[0]

Exercise 5. Using the same function, try sampling with the value of Omega set to double the true bandwidth (oversampling) and then half the true bandwidth (undersampling). How many samples are needed in the oversampling case to achieve a maximum pointwise error of 0.01 on the interval [-5,5]? Is it possible to achieve this accuracy in the undersampling case?

Exercise 6. Consider the Gaussian function Exp[-Pi t^2].  This function is not bandlimited, but, since the transform is also a Gaussian, we can choose an "effective bandwidth" resulting in a good representation.  Based on a graph of the transform, start with a reasonable value for Omega, and test how well the series from the Shannon sampling theorem reconstructs this function.  Experiment with different values and choose one that seems "optimal," explaining how you decided on that value (including your criteria for "optimal"). Don’t forget to add a constant term (k=0) to your sum.