Spectrum analysis using the discrete fourier transform  (Page 2/3)

Relationship between continuous-time ft and dft

The goal of spectrum analysis is often to determine the frequency content of an analog (continuous-time) signal; very often, as in most modern spectrum analyzers, this is actually accomplished by sampling the analog signal, windowing (truncating) the data, and computing and plotting the magnitude of its DFT.It is thus essential to relate the DFT frequency samples back to the original analog frequency. Assuming that the analog signal is bandlimited and the sampling frequency exceeds twice that limit so that no frequency aliasing occurs, the relationship betweenthe continuous-time Fourier frequency $\Omega$ (in radians) and the DTFT frequency $\omega$ imposed by sampling is $\omega =\Omega T$ where $T$ is the sampling period. Through the relationship ${\omega }_k=\frac{2\pi k}{N}$ between the DTFT frequency $\omega$ and the DFT frequency index $k$ , the correspondence between the DFT frequency index and the original analog frequency can be found: $$\Omega =\frac{2\pi k}{NT}$$ or in terms of analog frequency $f$ in Hertz (cycles per second rather than radians) $$f=\frac{k}{NT}$$ for $k$ in the range $k$ between $0$ and $\frac{N}{2}$ . It is important to note that $k\in \left[\frac{N}{2}+1 , N-1\right]()()$ correspond to negative frequencies due to the periodicity of the DTFT and the DFT.

Zero-padding

If more than $N$ equally spaced frequency samples of a length- $N$ signal are desired, they can easily be obtained by zero-padding the discrete-time signal and computing a DFT of the longer length.In particular, if $LN$ DTFT samples are desired of a length- $N$ sequence, one can compute the length- $LN$ DFT of a length- $LN$ zero-padded sequence $$z(n)=\begin{cases}x(n) & \text{if $0\le n\le N-1$}\\ 0 & \text{if $N\le n\le LN-1$}\end{cases}$$ $$X(w_k=\frac{2\pi k}{LN})=\sum_{n=0}^{N-1} x(n)e^{-(i\frac{2\pi kn}{LN})}=\sum_{n=0}^{LN-1} z(n)e^{-(i\frac{2\pi kn}{LN})}={\mathrm{DFT}}_{LN}(z(n))$$ Note that zero-padding interpolates the spectrum. One should always zero-pad (by about at least a factor of 4) whenusing the DFT to approximate the DTFT to get a clear picture of the DTFT . While performing computations on zeros may at first seem inefficient,using FFT algorithms, which generally expect the same number of input and output samples, actually makes thisapproach very efficient.

shows the magnitude of the DFT values corresponding to the non-negative frequencies of a real-valued length-64 DFT of a length-64 signal,both in a "stem" format to emphasize the discrete nature of the DFT frequency samples, and as a line plot to emphasize its use as an approximation to thecontinuous-in-frequency DTFT. From this figure, it appears that the signal has a single dominantfrequency component.