The sinx/x and the sinx/x squared response is a very popular response that occurs repeatedly in the literature in electrical engineering, mechanical engineering, economics, finance, medicine etc. The image shown here presents the sinx/x squared and the sinx/x response superimposed on each other. Note the difference in responses. For example if sinx/x is the amplitude response then sinx/x squared is the power response. !
It seems that DSP becomes confusing when you start, either reading about, or writing about, DSP quantities like filtering and modeling. Here is the scoop on this, to help if you need it.
First, Nyquist frequency and Nyquist rate: A band limited signal may be sampled by another signal (fs) that has a frequency two times the bandwidth ( B). B is called the Nyquist frequency and fs is called the Nyquist rate or simply the sample rate. Nyquist frequency is fs/2 if fs is twice B. These quantities can be normalized to fs = 1 ( normalized to fs) and the Nyquist frequency can be normalized to 0.5. ( Normalized to (fs/2)/fs.). Normalized Nyquist rate = 1. Normalized Nyquist frequency = 0.5. Samples/second. Some CAD tools use the Nyquist frequency as the normalization constant ( MATLAB filter design).
When the non normalized frequency is in terms of w, the angular frequency ( radians/second) and when this is normalized by the sample rate in samples/second the result is in radians/sample. Thus the normalized Nyquist frequency is pi radians/sample, and the normalized Nyquist rate is 2pi radians/sample.
There are also conversions available ( Wikipedia). f(radians/sample)=f(cycles/sec)*(2pi)/fs.
Decimation is a clever technique in digital signal processing that allows multi-rate processing of digital signals. A high sampling rate signal can be down sampled in such a manner that data is not lost in the process and yet a communication link is established. For example , a long wave communication channel for communicating with submarines. To further our understanding of decimation lets consider a band limited signal sampled at 300 Khz.
Assume the highest frequency component in the signal is at 100 Khz. If I want to reduce the sampling rate down to 100 Khz, then I must make sure there are no components of the input signal above 50 kHz. However, as mentioned above the signal has some components at 100 Khz. So if we really want a low sampling rate we will have to band limit the signal to 50 Khz. We must filter the signal before applying it to a down sampler. This technique and the combination of filter and down sampler is called DECIMATION.