Spectral Resolution and Zero-Padding

• We have seen that the DFT decomposes a signal into a harmonic set of sinusoidal amplitudes and phases at frequencies that are evenly distributed from 0 to $f_s\frac{N-1}{N},$ where $N$ is the DFT length and $f_s$ is the sample rate. Only the components less than or equal to $f_s/2$ are unique.

• The larger $N$, the more sinusoids will be available for comparison, thus providing better frequency resolution.

• At the same time, computations over larger values of $N$ “average out” time variations in the signal. If the frequency content of a signal is changing relatively quickly in time, smaller values of $N$ may be needed to discern those variations.

• This trade-off between time- and frequency-resolution is referred to as the uncertainty principle.

• It turns out that we can add zeros to the end of a signal in order to achieve a longer DFT length without modifying the spectral content of the signal. Because the zero-padded signal is longer (though no new energy has been added), the resulting DFT provides better frequency resolution.

• For example, a length $N=1024$ DFT provides a frequency resolution of 43 Hertz at a sample rate of 44.1 kHz. If we add $N$ more zeros to the signal and perform a length 2048 DFT, we double the frequency resolution (to 21.5 Hz).

• Stated another way, zero-padding in the time-domain results in interpolation in the frequency-domain.

• Figure 5 illustrates the effect of zero padding on the resulting DFT.

  Figure 5: A time-domain signal (top), its DFT (middle), and the DFT when zero-padded by a factor of 3 (bottom).