by Nasser M. Abbasi, added 2/27/2010
reference: lecture notes by Dr Shiva, EE 420 (digital filters), CSUF and DSP textbook, by Oppenheim, 1975, and textbook DSP by Proakis, 3rd edition.
I made these notes using diagrams, as I like to see things on diagram to help me understand them.
\(\Omega \) is radial frequency in rad/sec. Used for CTFT
\(\omega \) is radial frequency in radians. Used for DTFT
\(\omega =\Omega T\) where \(T\) is sampling frequency.
We start with \(f\left ( t\right ) \), the continuous time signal. We multiply it by a train of impulses called \(g\left ( t\right ) ={\sum \limits _{n=-\infty }^{\infty }}\delta \left ( t-nT\right ) \) where \(T\) is the sampling period. From this we obtain \(\tilde{f}\left ( t\right ) \) the sampled version of \(f\left ( t\right ) \). Now we can obtain \(f\left ( t\right ) \) by interpolation.
To see how to do the above in frequency domain, we start by taking CTFT of \(f\left ( t\right ) \) to obtain \(F\left ( \Omega \right ) \) and take the CTFT of \(g\left ( t\right ) \) to obtain \(G\left ( \Omega \right ) \)
Next, we need to map the operation of \(f\left ( t\right ) \times g\left ( t\right ) \) that we did in time domain to frequency domain. This becomes a convolution in frequency domain, but with \(\frac{1}{2\pi }\) at the front. Since \(G\left ( \Omega \right ) =\Omega _{s}{\sum \limits _{n=-\infty }^{\infty }}\delta \left ( \Omega -n\Omega _{s}\right ) \,,\)Hence we obtain\begin{align*} \tilde{G}\left ( \Omega \right ) & =\frac{1}{2\pi }\left [ F\left ( \Omega \right ) \circledast G\left ( \Omega \right ) \right ] \\ & =\frac{1}{2\pi }\left [ F\left ( \Omega \right ) \circledast \Omega _{s}{\sum \limits _{n=-\infty }^{\infty }}\delta \left ( \Omega -n\Omega _{s}\right ) \right ] \\ & =\frac{\Omega _{s}}{2\pi }{\sum \limits _{n=-\infty }^{\infty }}F\left ( \Omega -n\Omega _{s}\right ) \\ & =\frac{1}{T}{\sum \limits _{n=-\infty }^{\infty }}F\left ( \Omega -n\Omega _{s}\right ) \end{align*}
Where we used the convolution property that \(F\left ( \Omega \right ) \circledast{\sum \limits _{n=-\infty }^{\infty }}\delta \left ( \Omega -n\Omega _{s}\right ) ={\sum \limits _{n=-\infty }^{\infty }}F\left ( \Omega -n\Omega _{s}\right ) \), i.e convolving a function with a train of impulses, is the same function evaluated at the location of these impulses. We now see the effect of sampling. The original \(F\left ( \Omega \right ) \) is scaled by \(\frac{1}{T}\) and is duplicated every \(\Omega _{s}\)
Now we multiply \(\frac{1}{T}{\sum \limits _{n=-\infty }^{\infty }}F\left ( \Omega -n\Omega _{s}\right ) \) by a low pass filter \(T\ rect\left ( \frac{\Omega }{\Omega _{s}}\right ) \), which pulls out \(F\left ( \Omega \right ) \) out of those duplications and adjust the scale back. So now we have the original \(F\left ( \Omega \right ) \). But if we wanted to obtain \(f\left ( t\right ) \) back, then we need to convert the operation
\[ \frac{1}{T}{\sum \limits _{n=-\infty }^{\infty }}F\left ( \Omega -n\Omega _{s}\right ) \times T\ rect\left ( \frac{\Omega }{\Omega _{s}}\right ) \rightarrow F\left ( \Omega \right ) \]
to the time domain
\[ CTFT^{-1}\left ( \frac{1}{T}{\sum \limits _{n=-\infty }^{\infty }}F\left ( \Omega -n\Omega _{s}\right ) \right ) \circledast CTFT^{-1}\left ( T\ rect\left ( \frac{\Omega }{\Omega _{s}}\right ) \right ) \rightarrow f\left ( t\right ) \]
i.e. we apply inverse fourier transform and since multiplication become convolution, we need to convolve a sinc function (which is the inverse transform of the rect) with the inverse fourier transform of \(\frac{1}{T}{\sum \limits _{n=-\infty }^{\infty }}F\left ( \Omega -n\Omega _{s}\right ) \) which is \({\sum \limits _{n=-\infty }^{\infty }}f\left ( nT\right ) \delta \left ( t-nT\right ) \), hence
\begin{align*} f\left ( t\right ) & ={\sum \limits _{n=-\infty }^{\infty }}f\left ( nT\right ) \delta \left ( t-nT\right ) \circledast \operatorname{sinc}\left ( \frac{\Omega _{s}}{2}t\right ) \\ & ={\sum \limits _{n=-\infty }^{\infty }}f\left ( nT\right ) \operatorname{sinc}\left ( \frac{\Omega _{s}}{2}t-nT\right ) \end{align*}
Therefore
\[ f\left ( t\right ) ={\sum \limits _{n}}f\left ( nT\right ) \operatorname{sinc}\left ( \frac{\Omega _{s}}{2}t-nT\right ) \]
To help understand the above, these are some diagrams. This diagram show sampling in time domain. Then shows the same steps, but done in frequency domain.
This diagram shows the operation of obtaining the original continues signal from its samples using convolution with a sinc function, and how it came about.
