Truncation of the Infinite Fourier Transform - Signal Processing of Random Physiological Signals - page 158

where ε n is the truncation error. The partial sum is often expressed as the “mean-squared

error” as in (14.2).

T

1
T

ε n ( t )] 2 dt

E n = [ (14.2)

0

To understand truncation of a series, it is important to know what is desirable in
truncation. Since accuracy is one of the most desirable specifications in any application, it
would be desirable to attain acceptable accuracy with the least number of terms. Without
proof in limits and convergence, let us define convergence as the rate at which the
truncated series approaches the value of the original function, f ( t ). In general, the more
rapid the convergence, the fewer terms are required to obtain desired accuracy.
The convergence rate of the Fourier Transform is directly related to the rate of
the decrease in the magnitude of the Fourier Coefficients. Recall the Fourier transform
coefficients of a square wave as having the following magnitudes:

1
3

1
5

1
7

1
9

1
n

| a 3 | = ; | a 5 | = ; | a 7 | = ; | a 9 | = ; ···· ; | a n | =

Note that the sequence is the reciprocal of the n th term to the 1st power.
Now let us examine the Fourier transform of the triangular waveform shown in
Fig. 14.1.
The Fourier trigonometric series representation is given by (14.3).

sin

ω 0 t

8 V
π

1
3 2

1
5 2

1
n 2

v ( t ) = ω 0 t − sin 3 ω 0 t + sin 5 ω 0 t +···+ sin n (14.3)

2

Note that the magnitude of the coefficients are the square of the n th term.

1
n 2
It is noted that the Fourier coefficients for a triangular waveform diminish faster (rate
of convergence) than the Fourier coefficients for a rectangular waveform. To help in

1
3 2 ;

1
5 2 ;

1
7 2 ;

1
9 2 ;

| a 3 | = | a 5 | = | a 7 | = | a 9 | = ···· ; | a n | =

-T /2 -T /4 T /2

Odd

T /4

FIGURE 14.1 : One cycle of a triangular waveform with odd symmetry