In spectral analysis it is important to note that the discrete Fourier transform

(DFT) and its associated inverse (IDFT) can be represented in more than one

form. The method by which other functions of the DFT are computed depends

on an understanding of which DFT type was used. Presented here are two forms

of the DFT and an example of how the power-spectral-density function, by

example, is determined for each form. The first is labeled here as the DFT

proper since it is the derived equivalent of the continuous Fourier transform. The

second form, however, is more commonly implemented in practice (including

data analysis programs such as LabView or MATLAB), and thus is labeled here

as the algorithmic DFT.

DFT Proper:

The continuous Fourier transform for a given function h( t ) is given as

H ( f )=∫−∞

∞

h( t )e−i2 π ftdt

with the following inverse

h( t )=∫−∞

∞

H ( f )e i2 π ft df

A discrete and periodic form of these transformations are

Hm= Δt∑n=0

N−1

hn e−i 2π nm

N

hn=Δf ∑m=0

M−1

Hm ei 2 π

nmN

sincetn=nΔt and f m=mΔf=m /(NΔt ).

If h has the generic units of h and t is in seconds then H has the units of h/Hz.

[It can be argued that the unit 1/cycle can be correctly added to the transform.

Although no formal literature has been found to assert this, the concept can be

strengthened if the radian form is considered, e−iθ

. In this case the transform

and its inverse are not symmetric requiring a scaling factor of 2π to be

appropriated between the transforms. The arbitrary unit of radians can be

assigned, as allowable, and combined with the scaling factor to produce the unit

cycles. Since the transform should be invariant of the choosing of cycles or

radians (the exponential e has no units), the form as presented should also

include the unit cycles to make both versions (radian and cycle) of the transforms

compatible.] Since H is complex the units on the spectral density have little

applicable meaning; however it is important to note for the calculation of the

power spectral density which is real valued. The calculation of the power

spectral density function is given as

G=H ¿HΔf

with the units of h^2/Hz. The function H¿ is the complex conjugate of H .

The Algorithmic DFT:

For most applications the DFT has been defined to be invariant of a scaled time

and frequency. The algorithm then only requires information on the data series hn for calculation. By observing from before that

Δf=1/ (NΔt )

the discrete transform set can be rewritten by dividing the DFT by Δt . This also

requires multiplying the IDFT by Δt to maintain correct scaling.

Hm= ∑n=0

N−1

hn e−i 2 π nm

N

hn=1N

∑m=0

M−1

H mei2 π

nmN

In this form the units of Hm and hn are identical since no information regarding

time or frequency is conveyed. Hm is distinguished from Hm since the results of

these two functions are proportional, but not equal. The calculation of the power

spectral density however still requires frequency information, but because of the

redefinition of the transform it now becomes

G= H ¿ H

N2Δf

in order to maintain the correct units. Below is a MATLAB example illustrating

the calculation of a PSD for a sinusoidal oscillation at 64Hz and of amplitude 1. If

the units for this signal are volts/sqrt(Ohm) [eg. 1 volt acting across a 1 Ohm

resistor] then the expected total power output would be 0.5 watts [an equivalent

DC source with a magnitude of the RMS of the signal, or 1/sqrt(2)

volts/sqrt(Ohm), will produce the same power]. The power output is invariant to

the frequency. The plot of the PSD is multiplied by 2 since we are plotting only

the range below the Nyquist frequency, but still need to account for the power of

the mirroring side.

Fs=1000; % sample rate [samples/second]dt=1/Fs; % sample time [seconds]N=2048; % number of samplesT=N*dt; % sample period [seconds]t=linspace(0,T,N); % time vector [seconds]h=sin(2*pi*64*t); % signal [volts/sqrt(Ohm)] df=1/T; % frequency bin size [Hz]f=linspace(0,1/dt,N); % frequency vector [Hz] H=fft(h); % FFT of h [volts/sqrt(Ohm)/Hz] G=conj(H).*H/N/N/df; % PSD of h [(volts^2/Ohm)/Hz or Watts/Hz] TotalPower = sum(G)*df % Total Power [Watts] plot(f(1:(N/2-1)),2*G(1:(N/2-1))) % [Hz] [Watts/Hz]