Signals and Systems - Electrical Engineering

270 C H A P T E R 4: Frequency Analysis: The Fourier Series

nExample 4.11

Consider the train of pulses in Example 4.5. Determine how many Fourier coefficients are necessary

to get a representation containing 97% of the power of the periodic signal.

Solution

The desired 97% of the power ofx(t)is

0.97

1

T 0

∫

T 0

x^2 (t)dt=0.97

0.25∫

−0.25

4 dt=1.94

and so we need to find an integerNsuch that

∑N

k=−N

|Xk|^2 =

∑N

k=−N

∣

∣

∣

∣

sin(πk/ 2 )

(πk/ 2 )

∣

∣

∣

∣

2

=1.94

The value ofNis found by trial and error, adding consecutive values of the magnitude squared

of Fourier coefficients. Using MATLAB, it is found that forN=5 (dc and 5 harmonics) the

Fourier series approximation has a power of 1.93. Thus, 11 Fourier coefficients give a very good

approximation to the periodic train of pulses, with about 97% of the signal power. n

4.8 Time and Frequency Shifting................................................................

Time shifting and frequency shifting are duals of each other.

n Time-shifting: A periodic signalx(t), of periodT 0 , remains periodic of the same period when shifted in

time. IfXkare the Fourier coefficients ofx(t), the Fourier coefficients forx(t−t 0 )are

{

Xke−jk^0 t^0 =|Xk|ej(∠Xk−k^0 t^0 )

}

(4.27)

That is, only a change in phase is caused by the time shift. The magnitude spectrum remains the same.

n Frequency-shifting: When a periodic signalx(t), of periodT 0 , modulates a complex exponentialej^1 t:

n The modulated signalx(t)ej^1 tis periodic of periodT 0 if 1 =M 0 for an integerM≥ 1.

n The Fourier coefficientsXkare shifted to frequenciesk 0 + 1.

n The modulated signal is real-valued by multiplyingx(t)bycos( 1 t).

If we delay or advance in time a periodic signal, the resulting signal is periodic of the same period.

Only a change in the phase of the coefficients occurs to accommodate for the shift. Indeed, if

x(t)=

∑

k

Xkejk^0 t