0.6 Digital filtering and the dft  (Page 7/13)

With $\mathtt{N}=2^{16}$ , specgong.m analyzes approximately 1.5 seconds ( Ts*N seconds, to be precise). It is reasonable to suppose that the gong might undergo important transients duringthe first few milliseconds. This can be investigated by decreasing N and applying the DFT to different segments of the data record.

The key factors in a DFT or FFT based frequency analysis are as follows:

• The sampling interval $T_s$ is the time resolution, the shortest time over which any event can be observed.The sampling rate $f_s=\frac{1}{T_s}$ is inversely proportional.
• The total time is $T=N{T}_s$ where $N$ is the number of samples in the analysis.
• The frequency resolution is $\frac{1}{T}=\frac{1}{N{T}_s}=\frac{f_s}{N}$ . Sinusoids closer together (in frequency) than this value areindistinguishable.

For instance, in the analysis of the gong conducted in specgong.m , the sampling interval $T_s=\frac{1}{44100}$ is defined by the recording. With $N=2^{16}$ , the total time is $N{T}_s=1.48$ seconds, and the frequency resolution is $\frac{1}{N{T}_s}=0.67$ Hz.

Sometimes the total absolute time $T$ is fixed. Sampling faster decreases $T_s$ and increases $N$ , but cannot give better resolution in frequency.Sometimes it is possible to increase the total time. Assuming a fixed $T_s$ , this implies an increase in $N$ and better frequency resolution. Assuming a fixed $N$ , this implies an increase in $T_s$ and worse resolution in time. Thus, better resolution in frequency means worse resolution intime. Conversely, better resolution in time means worse resolution in frequency.

The DFT is a key tool in analyzing and understanding the behavior of communications systems. Whenever data flows through a system,it is a good idea to plot it as a function of time, and also to plot it as a function of frequency; that is, to look atit in the time domain and in the frequency domain. Often, aspects of the data that are clearer in time are hardto see in frequency, and aspects that are obvious in frequency are obscure in time. Using both points of view is common sense.