Signal Processing Magic (5) -- Windowing

In discrete-time signal processing, if received signal is a tone, its spectrum looks like a sinc function (see this). The reason is by cutting digital signal to finite length, it is like imposing a rectangular window in time domain. And rectangular window in time domain corresponds to sinc function in freq domain.

But some engineers thought side lobes in sinc function are too big. In order to make them smaller, they invented windowing method, which is to impose a window on the time-domain signal and it can suppress side lobes in the frequency domain. Since then people have invented different types of windows (Bartlett, Hanning, Hamming, Blackman etc. see Sec 7.2 in Discrete-time signal processing written by Oppenheim&Schafer). Hamming and Hanning windows are shown below. Assuming the digital signal has M samples, to apply windowing, one should generate a window lasting M samples and then multiple window with digital sample by sample. Each window is well defined mathematically. For example, Hamming window for M samples is defined as:
\[w[n]=0.54-0.46cos(2\pi n/M)\]

Next we compare two signals, one with windowing and one without. The signal is a 100KHz tone with sampling rate of 1MHz. The plot below shows that by adding Hamming window, the side lobe is suppressed by around 20dB but main lobe becomes wider. Windowing function works like a magical rolling pin. It drives the signal energy from side lobes to the main lobe.

In the next plot, instead of one tone, we show the spectrum of two tones, one at 100KHz and another at 300KHz. After applying windowing, these two tones become more distinctive.