<< Chapter < Page Chapter >> Page >

A signal's Fourier series spectrum c k has interesting properties.

Property

If s t is real, c k c k (real-valued periodic signals have conjugate-symmetric spectra).

This result follows from the integral that calculates the c k from the signal. Furthermore, this result means that c k c k : The real part of the Fourier coefficients for real-valuedsignals is even. Similarly, c k c k : The imaginary parts of the Fourier coefficients have oddsymmetry. Consequently, if you are given the Fourier coefficients for positive indices and zero and are told thesignal is real-valued, you can find the negative-indexed coefficients, hence the entire spectrum. This kind of symmetry, c k c k , is known as conjugate symmetry .

Property

If s t s t , which says the signal has even symmetry about the origin, c k c k .

Given the previous property for real-valued signals, the Fourier coefficients of even signals are real-valued. A real-valuedFourier expansion amounts to an expansion in terms of only cosines, which is the simplest example of an even signal.

Property

If s t s t , which says the signal has odd symmetry, c k c k .

Therefore, the Fourier coefficients are purely imaginary. The square wave is a great example of an odd-symmetric signal.

Property

The spectral coefficients for a periodic signal delayed by τ , s t τ , are c k 2 k τ T , where c k denotes the spectrum of s t . Delaying a signal by τ seconds results in a spectrum having a linear phase shift of 2 k τ T in comparison to the spectrum of the undelayed signal. Notethat the spectral magnitude is unaffected. Showing this property is easy.

1 T t 0 T s t τ 2 k t T 1 T t τ T τ s t 2 k t τ T 1 T 2 k τ T t τ T τ s t 2 k t T
Note that the range of integration extends over aperiod of the integrand. Consequently, it should not matter how we integrate over a period, which means that t τ T τ · t 0 T · , and we have our result.

The complex Fourier series obeys Parseval's Theorem , one of the most important results in signal analysis.This general mathematical result says you can calculate a signal's power in either the time domain or the frequencydomain.

Parseval's theorem

Average power calculated in the time domain equals the powercalculated in the frequency domain.

1 T t 0 T s t 2 k c k 2
This result is a (simpler) re-expression of how tocalculate a signal's power than with the real-valued Fourier series expression for power.

Let's calculate the Fourier coefficients of the periodic pulse signalshown here .

Periodic pulse signal.
The pulse width is Δ , the period T , and the amplitude A . The complex Fourier spectrum of this signal is given by c k 1 T t 0 Δ A 2 k t T A 2 k 2 k Δ T 1 At this point, simplifying this expression requires knowing aninteresting property. 1 θ θ 2 θ 2 θ 2 θ 2 2 θ 2 Armed with this result, we can simply express the Fourierseries coefficients for our pulse sequence.
c k A k Δ T k Δ T k
Because this signal is real-valued, we find that thecoefficients do indeed have conjugate symmetry: c k c k .The periodic pulse signal has neither even nor odd symmetry; consequently, no additional symmetry exists in the spectrum.Because the spectrum is complex valued, to plot it we need to calculate its magnitude and phase.
c k A k Δ T k
c k k Δ T neg k Δ T k sign k The function neg · equals -1 if its argument is negative and zero otherwise. The somewhat complicated expression for the phase resultsbecause the sine term can be negative; magnitudes must be positive, leaving the occasional negative values to be accountedfor as a phase shift of .

Periodic pulse sequence

The magnitude and phase of the periodic pulse sequence'sspectrum is shown for positive-frequency indices. Here Δ T 0.2 and A 1 .

Also note the presence of a linear phase term (the first term in c k is proportional to frequency k T ). Comparing this term with that predicted from delaying a signal,a delay of Δ 2 is present in our signal. Advancing the signal by this amountcenters the pulse about the origin, leaving an even signal, which in turn means that its spectrum is real-valued. Thus, ourcalculated spectrum is consistent with the properties of the Fourier spectrum.

What is the value of c 0 ? Recalling that this spectral coefficient corresponds to thesignal's average value, does your answer make sense?

c 0 A Δ T . This quantity clearly corresponds to the periodic pulsesignal's average value.

Got questions? Get instant answers now!

The phase plot shown in [link] requires some explanation as it does not seem to agree with what [link] suggests. There, the phase has a linear component, with a jump of every time the sinusoidal term changes sign. We must realize that any integer multiple of 2 can be added to a phase at each frequency without affecting the value of the complex spectrum. We see that at frequency index 4 the phase is nearly . The phase at index 5 is undefined because the magnitude is zeroin this example. At index 6, the formula suggests that the phase of the linear term should be less than (more negative). In addition, we expect a shift of in the phase between indices 4 and 6. Thus, the phase value predicted by the formula is a little less than 2 . Because we can add 2 without affecting the value of the spectrum at index 6, theresult is a slightly negative number as shown. Thus, the formula and the plot do agree. In phase calculations like those made inMATLAB, values are usually confined to the range by adding some (possibly negative) multiple of 2 to each phase value.

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Fundamentals of electrical engineering i. OpenStax CNX. Aug 06, 2008 Download for free at http://legacy.cnx.org/content/col10040/1.9
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Fundamentals of electrical engineering i' conversation and receive update notifications?

Ask