2.7  Advanced: Summary of equations for DFT / FFT Usage

The DFT resolution in radians is $\mathrm{\Delta }\mathrm{\Omega }=2\pi ∕N$. Using $\omega =\mathrm{\Omega }{F}_s$, one can note that the continuous-time angular frequency $\omega =2\pi F_s$ rad/s corresponds to the angle $\mathrm{\Omega }=2\pi$ rad. Hence, the DFT frequency spacing is

For example, assuming $F_s=100$ Hz, a DFT of $N=256$ points has a resolution of $\mathrm{\Delta }f=0.3906$ Hz. In radians, this resolution corresponds to $\mathrm{\Delta }\mathrm{\Omega }=2\pi ∕256\approx 0.0245$ rad.

Note that $\mathrm{\Delta }f$ can also be written as $\mathrm{\Delta }f=F_s∕N=1∕(N{T}_s)$, where $N{T}_s$ is the total duration $T_{\mathrm{\text{total}}}$ of the signal being analyzed, such that

Hence, a spectral analysis with resolution of at least $\mathrm{\Delta }f$ Hz requires a data record of length

The following example illustrates the interpretation in Hz of a DFT result. The DFT performs a sampling operation on the frequency domain $\mathrm{\Omega }$ according Eq. (2.35). As illustrated in Figure 2.7, a DFT with $N=3$ points (or 3-DFT) uses three basis functions $e^{j\mathrm{\Omega }}$ with the angles $\mathrm{\Omega }=0,2\pi ∕3,4\pi ∕3$ rad, i. e., a counter-clockwise rotation using a step of $\mathrm{\Delta }w=2\pi ∕3$. These angles correspond to $k=0,1,2$, respectively, as indicated by Eq. (2.11). Note that in Figure 2.7, incrementing $k$ corresponds to a clockwise rotation, given that Eq. (2.17) defines the twiddle factor with a negative exponent.

The three angles of a 3-DFT correspond to $0,F_s∕3,-F_s∕3$ Hz, respectively, as indicated by $\omega =\mathrm{\Omega }{F}_s$ or directly observing that the DFT uses a grid of $\mathrm{\Delta }f=F_s∕N$. For $N=4$, the four angles correspond to the frequencies $0,F_s∕4,F_s∕2,-F_s∕4$ Hz. The angle corresponding to $k=0$ is always 0 rad, which is called DC because corresponds to 0 Hz. Note that for $k>N∕2$ the DFT values correspond to negative frequencies. The angular frequency corresponding to the $k$-th angle is

which corresponds to the regular (linear) frequency

The values representing the largest frequency when $N$ is even is

which corresponds to $\pi$ rad and $F_s∕2$ Hz. When $N$ is odd, the value at

is the one representing the largest frequency $F_s(N-1)∕(2N)$ Hz. Table 2.6 summarizes some equations typically used with FFT algorithms.

2.7.1  Advanced: Three normalization options for DFT / FFT pairs

As discussed, the following three transforms differ only on the normalization factor: DFT, unitary DFT and DTFS. Generalizing Eq. (2.20) and Eq. (2.21), the “core” DFT transform equations can be written as follows:

and

As indicated in Eq. (2.25), the only requirement to have a valid pair is that

In summary, the three popular options for choosing $\alpha$ and $\beta$:

• DFT as samples of the DTFT (in frequency domain):

  $$\alpha =1\mathrm{\text{ and }}\beta =\frac{1}{N}.$$

  (2.47)

  Matlab/Octave implements this transform in its fft and ifft routines. It allows to estimate the DTFT at specific values of $\mathrm{\Omega }$ using Eq. (2.35). Also, it is the fastest in the direct transform because there is no scaling factor to be accounted for.

• Unitary DFT:

  $$\alpha =\beta =\frac{1}{\sqrt{N}}.$$

  (2.48)

  Because this transform is unitary, Theorem 2 applies and the energy sum(abs(X).^2) in frequency and time sum(abs(x).^2) domains coincide. To implement the unitary DFT when using Matlab/Octave one can call X=fft(x)/sqrt(N) and x=ifft(X)*sqrt(N) for the direct and inverse transforms, respectively.

• DFT as DTFS in case the number of DFT points coincides with the signal period:

  $$\alpha =\frac{1}{N}\mathrm{\text{ and }}\beta =1.$$

  (2.49)

  In this case, the DFT coefficients can be interpreted in volts when the time-domain signal is given in volts. If the signal is periodic with fundamental period $N_0=N$ coinciding with the number $N$ of DFT points, choosing Eq. (2.49) allows to obtain the DTFS coefficients via the DFT. To implement this version when using Matlab/Octave, one can call X=fft(x)/N and x=ifft(X)*N for the direct and inverse transforms, respectively.

The next two sections present the Laplace and Z transforms.¹⁴ Both have two parameters that identifies each basis functions, in contrast to Fourier equations that have only the frequency. This extra degree of freedom, represented by sets of basis functions that are more powerful than the ones used in Fourier analysis, has the advantage of allowing the representation of a larger class of signals (and systems). A disadvantage is that the inverse transform is defined as a contour integration in the complex plane, which is an area of complex analysis that is out of the scope of this text. But it is rarely necessary to calculate the inverse transform using complex analysis. The Laplace and Z inverse transforms will be calculated here only by an alternative method called partial fraction expansion, which is discussed in the Appendix, Section A.10.

The Laplace and Z transforms are widely used to represent the action of linear and time-invariant systems, as will be detailed in Chapter 3. The Laplace transform has a number of properties that make it useful for analyzing linear systems. The most prominent advantage is that differentiation and integration become multiplication and division, respectively, by $s$. For example, this converts differential equations into polynomial equations, which are much easier to solve and, once solved, the inverse Laplace transform can provide the time domain formula. Similarly, Z transforms are useful to analyze discrete-time systems.