The calculation of Fourier transforms on a digital computer requires the numerical evaluation of

F(ja>) = (1/T) j" x(t)e~i0,,dt (4.3.1)

Jo

or

Re{T(yco)} = (1/T) j" x(t) cos tot dt Jo

lm{F(ja>)} = -(1/T) f x(t) sin cot dt Jo

Any method for approximating the integral using data obtained at discrete sampling points is called a discrete Fourier transform (DFT). The most common method for calculating the DFT is to approximate the integral as

follows. (Other methods are discussed in Section 4.5.’

x(t)exp(-jcot) dt ^ X xp exP( At/T (4.3.4)

p = 0

where xp is the pth sample of x(t), N the number of samples, and At the time interval between samples. Since N = Т/At, we write

F'(jco) = (l/N) £ xp exp( —jo)tp

P= o

where F'(j(o) is the DFT approximation to F(jco).

Since the DFT is only an approximation to the true Fourier transform, it is necessary to evaluate the consequences of discrete sampling. After a signal is sampled, it is not possible to detect a component with frequency above 1/2АГ cycles/time or n/At radians/time (see Fig. 4.4). This frequency is called

Fig. 4.4. Effect of sampling on the detection of higher frequencies.

the Nyquist frequency. If the signal contains components above the Nyquist frequency, these higher-frequency components not only go undetected, they also contaminate the lower-frequency results. Since these higher-frequency results give apparent results for “names” (frequencies) different from their own, this effect is called aliasing.

The aliasing effect will be analyzed for the DFT approximation of Eq. (4.3.5) for a periodic signal

Fk = 0/ЛО Y xp exp( — M*p) (4.3.6)

P = 0

This may be written

CT

x(£) exp( — jcokt) d(t — tp) dt 0 ,

A	A A	A A
A
time

Fig. 4.5. The periodic delta function.

where d(t — tp) is the delta function. The delta function (see Fig. 4.5) may be written as a Fourier series (without approximation)

N — 1 oo

X S(t — tp)= X Dm exp(jcoj) (4.3.8)

p = 0 m = — oo

where u>m = 2mn/At. The evaluation of Dm gives

Dm = 1/At (4.3.9)

This series may be substituted into Eq. (4.3.7) to give

oo і» T

Fk’= X (1/Л x(t)exp[—j(a)k — wm)t] dt (4.3.10)

m = — oo * 0

The integral may be interpreted if we identify the following:

wk — wm = {2kn/T) — (2mn/At) = (2n/T)(k — mN) (4.3.11) and

CT

x(t)exp[-j{k — mN)(2n/T)t]dt = Fk_mN

0

Then we obtain

00

Fk = X F)c-miv

m = — oo

or

Fk — Fk + Fk_N 4- Fk + N + Fk_2N + Fk + 2N + ••• (4.3.14)

This shows that the DFT result at harmonic к is contaminated by higher harmonics.

It is useful to convert these results into the trigonometric form of the Fourier integral in order to interpret the results. Equations (2.5.9) and

(2.5.10) give

oo

Ak = A + X (AmN + k + AmN-k) (4.3.15)

m = 1

oo

Вк = В к + X (BmN + k ~ ^mN-k)

m = 1

where

Ak = (2 /Т)	x(t) cos cokt dt ‘ 0	(4.3.17)
Bk = (2 /Т)	x(t) sin cokt dt ‘0	(4.3.18)

Thus the weighting of the influence of the higher harmonics is as shown in Fig. 4.6. The first alias frequency is at N — к (m = 1). The frequency corre­sponding to harmonic N is 2л/At. This is recognized as twice the Nyquist frequency. Clearly, the DFT will be accurate only if the contributions from

к N-k N N + k 2N-k2N2N + k 3N-k3N3N + k

Harmonic

Fig. 4.6. Aliasing frequencies for analysis at harmonic k, for a sampling rate N.

the alias frequencies are small. There is nothing that can be done about aliasing after the signal is digitized. Thus the tester must make sure that there is insignificant energy in alias frequencies relative to the analysis fre­quencies. If the combination of sampling rate and frequency response of the system being tested accomplish this, then no additional action is required.

If not, then either the sampling rate must be increased or the high-frequency part of the signal must be suppressed by low-pass filtering prior to sampling. The low-pass filtering must be set to pass the frequencies of interest and to attenuate alias frequencies as shown in Fig. 4.7.