The Sampling Theorem states that you can completely reconstruct a continuous-time signal from discrete equally spaced samples if the highest frequency in the time signal is less than half the sampling frequency. Half the sampling frequency equals the Nyquist frequency. This theorem bridges the gap between continuous time signals and digital time signals. When you digitize a time signal in practical applications, side effects occur even when the data meets the Nyquist criterion. One of the most common side effects is energy leakage caused by the finite observation window.
When you use the FFT or DFT to measure the frequency content of your data, these transforms assume that the finite window of data is one period of a periodic signal. The observation window, then, can cause sharp transition changes to be introduced into your measured data.
You can minimize the effects of these transition edges by applying smoothing windows. You can think of these windows, which modify the spectral contents of the digitized waveform, as filtering operations. The type of window you should use depends upon your application requirements. The following example demonstrates the windowed and non-windowed spectrums of a signal composed of the sum of two sinusoids. The two sinusoids have amplitudes and frequencies, measured in cycles, as shown in the following illustration.
The block diagram for this example, shown in the following illustration, demonstrates how to use smoothing windows to reduce spectral leakage.
The following graph displays the results. The dashed line represents the spectrum of the digitized signal with no window applied. The solid line represents the windowed spectrum. Notice how the nonwindowed spectrum shows leakage that is more than 20 dB greater than the smaller sinusoid.
You can apply more sophisticated techniques to get a more accurate description of the original time-continuous signal in the frequency domain. However, in most applications, applying a smoothing window is sufficient to obtain a better frequency representation of the signal.