Sampling Theorem: How to know the value between two samples of a Signal

Question

According to Sampling theorem, in order to reconstruct a signal we need to sample it at the rate => twice the highest frequency component of that signal. (provided signal is band limited).

Let's say, we have a signal with f= 2MHz (highest freq component), so we will be sampling it at 4MHz or more as Sampling theorem say's.. and we will have N Samples in signal.

Now, what if I want to know the exact value of a sample (information) between two samples..??

How can we find the value at every possible instantaneous.. in a signal..when we know we only have N samples after sampling..??

I know we can increasing sampling rate to have more sample. But is there any other way to do it..?

This question was asked to me in an interview at NCRA-TIFR, Pune, India.

Answer 1

Interpolation using a Sinc function kernel. This assumes that the sampled signal was perfectly bandlimited to below half the sampling rate.

Note that perfectly bandlimited signal are infinite in extent. For finite-length "real world" signals, using a windowed Sinc interpolation kernel (thus a finite computation with a noise floor) is a common method to approximate points between samples (for upsampling, for instance).

Answer 2

Another way to do it (eg: converting N samples to 2N): take the N point FFT of the signal, then zero pad the frequency domain result to 2N. Then compute the 2N sized inverse.

But note that when zero padding the N point frequency domain result, the zero padding is done in the middle of the spectrum, and the Fs/2 bin is split between the upper and lower halves of the spectrum. For instance, using 8 points as an example, where element A is the zero frequency bin, and element E is your Fs/2 bin:

A B C D E F G H ---- zero pad ----> A B C D E/2 0 0 0 0 0 0 0 E/2 F G H -------> inverse FFT

Such an operation will give you a 2N result where every other value is an interpolated one.

Answer 3

watch Digital show & Tell from Xiph.org. Same situation only sound frequency. They explain better as i can why there is just one possible solution from every set of samples

Answer 4

Another practical approach is to delay the signal a fraction of the sample period using a fractional delay filter.