Contents
SAMPLING PART 1
Basic Definition of Sampling
Sampling is the extraction of values of a continuous signal at fixed intervals. We learn more about the frequency spectrum of a signal the faster we sample it. Naturally, if the signal changes much faster than the sampling rate, these changes will not be captured accurately and aliasing occurs.
Nyquist Sampling Theorem
The Nyquist Sampling theorem says that in order to capture all the frequency information of a bandlimited signal, the sampling frequency must be twice the maximum frequency of the signal. In other words, each frequency component must be sampled at least twice per period.
<insert nyquist sampling rate conditions here>
The Sampling Process
In theory, here is how we would like to sample our signals.
Step 1: Begin with a continuous function x(t).
Step 2: Sample x(t) using an impulse generator or comb function.
Step 3: Discretize the signal.
After Step 3, the signal is ready to be put through a discrete filter.
It is important to note that this is an idealization of the sampling process. To adhere to the Nyquist sampling theorem, the sampling frequency must be at least twice the maximum frequency. Often, we do not know what the maximum frequency of the signal is. To prevent the effects of aliasing, the signal is first put through a lowpass filter. This allows us to base the sampling frequency off of the cutoff frequency of the filter. This will reduce the effects of aliasing, but may also distort the signal, since higher frequencies are inevitably lost. We also cannot generate an impulse in real life. The actual methods used to sample a continuous time signal will be introduced in sampling part 2. Finally, a sampled signal must be quanitized before discretization. This is because digital filters are limited in what numbers they can represent. This depends on the number of bits your computer is based off of.
To get a better understanding of what is actually happening between Steps 1-3, it is good to observe the frequency domain representation of the signal as it passes through each stage of the sampling process. The following explanation adheres to the idealization of the sampling process.
From a Frequency Standpoint
Step 1: The signal x(t) may be periodic or aperiodic. If the signal is periodic, the frequency domain representation is discrete. If the signal is aperiodic, the frequency domain representation is continuous. A good way to remember this is to remember that sampling in time is convolving the frequency domain representation of your signal with an impulse train in the frequency domain. Conversely, sampling in the frequency domain is equivalent to convolving the time domain representation of your signal with an impulse train.
Step 2: When the signal x(t) is multiplied by the dirac comb p(t), this is equivalent to convolving the frequency domain representation of x(t) with the frequency domain representation of p(t). Since the Fourier Transform of the comb is also an impulse train in the frequency domain, the convolution of X(f) with P(f) simply makes copies of X(f) at each impulse with the magnitude of X(f) scaled by the sampling frequency. The sampled signal now has a frequency domain representation which is periodic with respect to the sampling frequency.
<insert math stuff to show this>
Step 3: To discretize the sampled signal, the frequency must be scaled such that the frequency is periodic with respect to 2*pi. This is because discrete time filters are periodic with respect to 2*pi.
<insert proof for discrete signals being periodic wrt 2*pi>
To transform the continuous time sampled signal the its discrete time representation, let f_d = [2*pi*f_c - 2*pi*k]/f_s (for all k = integer), where f_s is the sampling frequency and f_c is a frequency corresponding to the continuous time frequency domain representation. f_d is the corresponding frequency for the discrete time representation of the sampled signal. Since the sampled signal is periodic in the frequency domain, the 2*pi*k term is there to account for this.
Now that the signal has been discretized, a discrete time filter may be applied to it.
Reconstructing the Signal
After the signal has been sampled, discretized, and processed in discrete time, the signal must be reconstructed. The discrete time signal is periodic with respect to 2*pi. To place it back into context with the continuous time domain, the frequency must be scaled again such that the frequency domain is periodic with respect to the sampling frequency. To do this, let f_c = [f_d*f_s - f_s*k]/2*pi.
