0
$\begingroup$

Let

$$f(t)=\int_{0}^{t}\exp\left(-\frac{1}{1-s^{2}}\right)\,ds,\qquad|t|<1,\text{ otherwise }0.$$

Then the Shannon interpolation theorem gives a reconstruction formula for a sampled version of $f$. Namely, for appropriately chosen $T$:

$$f_{n}(t)=\sum_{n=-\infty}^{\infty}\int_{0}^{n\times 1/T}\exp\left(-\frac{1}{1-s^{2}}\right)\,ds\cdot\frac{\sin(T\times t-n)}{T\times t-n},\qquad |s|<1.$$

Suppose that $T=102$, then $f_{n}(t)$ should return a perfect reconstruction of $f(t)$ (provided that $102>2\times b$, where $\hat{f}(\xi)=0$ for $|\xi|>b$).

Now I tried to plot this on Matlab, but with no success. My most recent attempt was using:

x = -2:0.01:2 % Position vector Fs = 100; % Sampling frequency T = 1/Fs; % Sampling period L = 100; % Length of signal t = (0:L-1)*T; % Time vector s = @(x)heaviside(x+1).*heaviside(-x)+heaviside(x).*heaviside(1-x); % Step function phi = @(x) exp(-1./(1-x.^2)).*s(x); % Bump function f = @(t) symsum(integral(@(x)phi(x),0,n/102)*sinc(102*n-t),n,[-200 200]); plot(f,[0,1])

  • 1
    You should start with the discrete case : start from a **discrete signal** $x(n)$ whose DFT $X(f) = \sum_n x(n) e^{-2i \pi n f} , f \in [-\frac{1}{2},\frac{1}{2}]$ is supported on $[-\frac{1}{2M},\frac{1}{2M}]$, down-sample it to get $y(n) = x(n M)$, and apply the Shannon reconstruction theorem : let $\tilde{x}(n) = M\, y(n/M)$ if $M|n$ ($\tilde{x}(n) = 0$ otherwise), plot its DFT !! finally let $\hat{x}(n) = \tilde{x} \ast h(n)$ where $h$ is an **almost** ideal low-pass filter with appropriate frequency cut-off $f_c = 1/M$ – 2017-01-20
  • 0
    @user1952009 I'm slightly confused. Why is $f\in[-\frac{1}{2},\frac{1}{2}]$ and doesn't the Shannon reconstruction theorem involve a sinc interpolation? – 2017-01-23
  • 0
    You should plot the DFT (FFT) of some discrete signals. $f \in [-1/2,1/2]$ because $X(f+1) = X(f)$. Yes theoretically the Shannon reconstruction theorem involves the $sinc$ an ideal filter with infinite support, so in real life you have to choose a [finite approximation of it](https://en.wikipedia.org/wiki/Low-pass_filter#Ideal_and_real_filters) – 2017-01-23
  • 0
    @user1952009 But the idea of the reconstruction theorem is to recover $x(t)$, not $\hat{x}(n)$ which is what your reconstruction theorem appears to yield. – 2017-01-23
  • 0
    No. Try what I said : start from a discrete signal, downsample and recover it from the interpolation theorem. – 2017-01-23
  • 0
    Okay, so if I understood correctly; in my case, we have $x[n]$ with DFT given by $X(f)=\sum_{n}x[n]e^{-2\pi in f}$, $f\in[-1/2,1/2]$, supported on $[-1/2M,1/2M]$. Down-sample to get $y[n]=x[nM]$. We let $$\widetilde{x}[n]=\begin{cases}My[n/M],& M|n,\\0,&\text{otherwise}\end{cases}$$ Then its DFT is given by $$\begin{aligned}\widetilde{X}(f)&=\sum_{n\in\mathbb{Z}}\widetilde{x}[n]e^{-2\pi inf}\\&=\begin{cases}M\sum_{n\in\mathbb{Z}}y[n/M]e^{-2\pi inf},& M|n,\\0,&\text{otherwise}\end{cases}\\&=\begin{cases}M\sum_{n}x[n]e^{-2\pi inf},&M|n,\\0,&\text{otherwise}.\end{cases}\end{aligned}$$ – 2017-01-23
  • 0
    Yes but you need to make it real on matlab !! – 2017-01-23
  • 0
    @user1952009 Okay, so to plot on Matlab, I need to choose an $M$. I set $M=9$ so that $f_{s}>2B=2\cdot 1/2M=1/M=1/9$. Choose $f_{s}=1/M-1=1/8$ so that $T=8$. Then I have to plot $$\widetilde{D}(f)=\begin{cases}72\sum_{n=-\infty}^{\infty}\int_{0}^{8n}\exp\left(-\frac{1}{1-s^2}\right)\,ds\cdot e^{-2\pi inf},&\text{if }9|n,\text{ and }|n|<8\\ 0,&\text{otherwise}.\end{cases}$$ But as per usual, I am having real problems with Matlab... – 2017-01-23
  • 0
    Certainly, I can start with `x = -2:0.01:2 % Position vector Fs = 9; % Sampling frequency T = 1/Fs; % Sampling period L = 10; % Length of signal t = (0:L-1)*T; % Time vector s = @(x)heaviside(x+8).*heaviside(-x)+heaviside(x).*heaviside(8-x); % Step function r = @(x) exp(-1./(1-x.^2)).*s(x); % Bump function f = zeros(size(t)); for i = 1:length(t) f(i) = 72*integral(r,0,t(i)); end` However, I am having trouble with defining my `n` for `fft(f,n)` and also how to implement the piecewise aspect whereby the DFT is nonzero whenever $9|n$. – 2017-01-23

0 Answers 0