2.5 POWER SPECTRAL DENSITY AND CORRELATION

Recalling the definition of energy spectral density, Equation(2.73), we see that it is of use only for energy signals for which the integral of𝐺(𝑓)over all frequencies gives total energy, a finite quantity. For power signals, it is meaningful to speak in terms ofpower spectral density.

Analogous to𝐺(𝑓), we define the power spectral density𝑆(𝑓)of a signal𝑥(𝑡)as a real, even,

that is,

𝑃 =∫

∞

−∞𝑆(𝑓) 𝑑𝑓 =⟨ 𝑥2(𝑡)⟩

(2.139) where⟨

𝑥2(𝑡)⟩

= lim𝑇→∞ 1

2𝑇 ∫−𝑇𝑇 𝑥2(𝑡) 𝑑𝑡denotes the time average of𝑥2(𝑡). Since𝑆(𝑓)is a function that gives the variation of density of power with frequency, we conclude that it must consist of a series of impulses for the periodic power signals that we have so far considered.

Later, in Chapter 7, we will consider power spectra of random signals.

EXAMPLE 2.18

Considering the cosinusoidal signal

𝑥(𝑡) = 𝐴 cos(2𝜋𝑓0𝑡 + 𝜃) (2.140)

we note that its average power per ohm,12𝐴2, is concentrated at the single frequency𝑓0hertz. However, since the power spectral density must be an even function of frequency, we split this power equally between+𝑓0and−𝑓0hertz. Thus, the power spectral density of𝑥(𝑡)is, from intuition, given by

𝑆(𝑓) =1

4 𝐴2𝛿(𝑓 − 𝑓0) + 1 4 𝐴2𝛿(

𝑓 + 𝑓0

) (2.141)

Checking this by using(2.139), we see that integration over all frequencies results in the average power per ohm of1

2𝐴2.

■

2.5.1 The Time-Average Autocorrelation Function

To introduce the time-average autocorrelation function, we return to the energy spectral density of an energy signal,(2.73). Without any apparent reason, suppose we take the inverse Fourier transform of𝐺(𝑓), letting the independent variable be𝜏:

𝜙(𝜏)≜ℑ−1[𝐺(𝑓)] =ℑ−1[𝑋(𝑓)𝑋∗(𝑓)]

=ℑ−1[𝑋(𝑓)] ∗ℑ−1[𝑋∗(𝑓)] (2.142) The last step follows by application of the convolution theorem. Applying the time-reversal theorem (Item 3b in Table F.6 in Appendix F) to writeℑ−1[𝑋∗(𝑓)] = 𝑥(−𝜏)and then the convolution theorem, we obtain

𝜙(𝜏) = 𝑥(𝜏) ∗ 𝑥(−𝜏) =∫

∞

−∞𝑥(𝜆)𝑥(𝜆 + 𝜏) 𝑑𝜆

= lim

𝑇→∞∫

𝑇

−𝑇 𝑥(𝜆)𝑥(𝜆 + 𝜏) 𝑑𝜆(energy signal) (2.143) Equation(2.143)will be referred to as thetime-average autocorrelation function for energy signals. We see that it gives a measure of the similarity, or coherence, between a signal and a delayed version of the signal. Note that𝜙(0) = 𝐸, the signal energy. Also note the similarity of the correlation operation to convolution. The major point of(2.142)is that the autocorrelation function and energy spectral density are Fourier-transform pairs. We forgo further discussion

of the time-average autocorrelation function for energy signals in favor of analogous results for power signals.

The time-average autocorrelation function𝑅(𝜏)of a power signal𝑥(𝑡)is defined as the time average

𝑅 (𝜏) =⟨𝑥(𝑡)𝑥(𝑡 + 𝜏)⟩

≜ lim

𝑇→∞

1 2𝑇 ∫

𝑇

−𝑇𝑥(𝑡)𝑥(𝑡 + 𝜏) 𝑑𝑡(power signal) (2.144) If𝑥(𝑡)is periodic with period𝑇0, the integrand of(2.144)is periodic, and the time average can be taken over a single period:

𝑅(𝜏) = 1

𝑇0∫𝑇0𝑥(𝑡)𝑥(𝑡 + 𝜏) 𝑑𝑡 [𝑥(𝑡)periodic]

Just like𝜙(𝜏), 𝑅(𝜏)gives a measure of the similarity between a power signal at time𝑡and at time𝑡 + 𝜏; it is a function of the delay variable𝜏, since time,𝑡, is the variable of integration.

In addition to being a measure of the similarity between a signal and its time displacement, we note that the total average power of the signal is

𝑅(0) =⟨ 𝑥2(𝑡)⟩

=∫

∞

−∞𝑆(𝑓) 𝑑𝑓 (2.145)

Thus, we suspect that the time-average autocorrelation function and power spectral density of a power signal are closely related, just as they are for energy signals. This relationship is stated formally by theWiener--Khinchine theorem, which says that the time-average autocorrelation function of a signal and its power spectral density are Fourier-transform pairs:

𝑆(𝑓) =ℑ[𝑅(𝜏)] =∫

∞

−∞𝑅 (𝜏) 𝑒−𝑗2𝜋𝑓𝜏𝑑𝜏 (2.146) and

𝑅 (𝜏) =ℑ−1[𝑆(𝑓)] =∫

∞

−∞𝑆(𝑓)𝑒𝑗2𝜋𝑓𝜏𝑑𝑓 (2.147)

A formal proof of the Wiener--Khinchine theorem will be given in Chapter 7. We simply take(2.146)as the definition of power spectral density at this point. We note that(2.145) follows immediately from(2.147)by setting𝜏 = 0.

2.5.2 Properties ofR(𝜏)

The time-average autocorrelation function has several useful properties, which are listed below:

1. 𝑅(0) =⟨ 𝑥2(𝑡)⟩

≥|𝑅 (𝜏)|, for all𝜏; that is, an absolute maximum of𝑅 (𝜏)exists at𝜏 = 0. 2. 𝑅 (−𝜏) =⟨𝑥(𝑡)𝑥(𝑡 − 𝜏)⟩= 𝑅 (𝜏); that is,𝑅(𝜏)is even.

3. lim|𝜏|→∞𝑅 (𝜏) =⟨𝑥(𝑡)⟩2if𝑥(𝑡)does not contain periodic components.

4. If𝑥(𝑡)is periodic in𝑡with period𝑇0, then𝑅 (𝜏)is periodic in𝜏with period𝑇0.

5. The time-average autocorrelation function of any power signal has a Fourier transform that is nonnegative.

Property 5 results by virtue of the fact that normalized power is a nonnegative quantity.

These properties will be proved in Chapter 7.

The autocorrelation function and power spectral density are important tools for systems analysis involving random signals.

EXAMPLE 2.19

We desire the autocorrelation function and power spectral density of the signal 𝑥 (𝑡) = Re[2 + 3 exp(𝑗10𝜋𝑡) + 4𝑗 exp(𝑗10𝜋𝑡)]or𝑥 (𝑡) = 2 + 3 cos (10𝜋𝑡) − 4 sin (10𝜋𝑡). The first step is to write the signal as a constant plus a single sinusoid. To do so, we note that

𝑥 (𝑡) = Re [2 +√

32+ 42exp[

𝑗 tan−1(4∕3)]

exp (𝑗10𝜋𝑡)]

= 2 + 5 cos[

10𝜋𝑡 + tan−1(4∕3)] We may proceed in one of two ways. The first is to find the autocorrelation function of𝑥 (𝑡)and Fourier-transform it to get the power spectral density. The second is to write down the power spectral density and inverse Fourier-transform it to get the autocorrelation function.

Following the first method, we find the autocorrelation function:

𝑅(𝜏) = 1

𝑇0∫𝑇0𝑥(𝑡)𝑥(𝑡 + 𝜏) 𝑑𝑡

= 10.2∫

0.2 0

{2 + 5 cos[

10𝜋𝑡 + tan−1(4∕3)]} {

2 + 5 cos[

10𝜋 (𝑡 + 𝜏) + tan−1(4∕3)]}

𝑑𝑡

= 5∫

0.2 0

{4 + 10 cos[

10𝜋𝑡 + tan−1(4∕3)]

+ 10 cos[

10𝜋 (𝑡 + 𝜏) + tan−1(4∕3)] +25 cos[

10𝜋𝑡 + tan−1(4∕3)] cos[

10𝜋 (𝑡 + 𝜏) + tan−1(4∕3)] }

𝑑𝑡

= 5∫

0.2

0 4𝑑𝑡 + 50∫

0.2 0

cos[

10𝜋𝑡 + tan−1(4∕3)] 𝑑𝑡

+50∫

0.2 0

cos[

10𝜋 (𝑡 + 𝜏) + tan−1(4∕3)] 𝑑𝑡

+ 125 2 ∫

0.2

0 cos (10𝜋𝜏) 𝑑𝑡 +125 2 ∫

0.2 0 cos[

20𝜋𝑡 + 10𝜋𝜏 + 2 tan−1(4∕3)] 𝑑𝑡

= 5∫

0.2

0 4𝑑𝑡 + 0 + 0 +125 2 ∫

0.2

0 cos (10𝜋𝜏) 𝑑𝑡 + 125

2 ∫

0.2 0

cos[

20𝜋𝑡 + 10𝜋𝜏 + 2 tan−1(4∕3)] 𝑑𝑡

= 4 + 25

2 cos (10𝜋𝜏) (2.148)

where integrals involving cosines of𝑡are zero by virtue of integrating a cosine over an integer number of periods, and the trigonometric relationshipcos 𝑥 cos 𝑦 =12cos (𝑥 + 𝑦) +12cos (𝑥 − 𝑦)has been used.

The power spectral density is the Fourier transform of the autocorrelation function, or 𝑆(𝑓) =ℑ[

4 + 25

2 cos (10𝜋𝜏)]

= 4ℑ[1] + 25

2ℑ[cos (10𝜋𝜏)]

= 4𝛿 (𝑓) +25

4 𝛿(𝑓 − 5) +25

4 𝛿(𝑓 + 5) (2.149)

Note that integration of this over all𝑓 gives𝑃 = 4 +252 = 16.5watts/ohm, which is the DC power plus the AC power (the latter is split between5and−5hertz). We could have proceeded by writing down the power spectral density first, using power arguments, and inverse Fourier-transforming it to get the autocorrelation function.

Note that all properties of the autocorrelation function are satisfied except the third, which does not apply.

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EXAMPLE 2.20

The sequence 1110010 is an example of a pseudonoise or m-sequence; they are important in the implementation of digital communication systems and will be discussed further in Chapter 9. For now, we use thism-sequence as another illustration for computing autocorrelation functions and power spectra. Consider Figure 2.11(a), which shows the waveform equivalent of thism-sequence obtained by replacing each 0 by−1, multiplying each sequence member by a square pulse functionΠ(

𝑡−𝑡0 Δ

) , summing, and assuming the resulting waveform is repeated forever thereby making it periodic. To compute the autocorrelation function, we apply(2.145), which is

𝑅(𝜏) = 1

𝑇0∫𝑇0𝑥(𝑡)𝑥(𝑡 + 𝜏) 𝑑𝑡

since a periodic repetition of the waveform is assumed. Consider the waveform 𝑥 (𝑡)multiplied by 𝑥 (𝑡 + 𝑛Δ)[shown in Figure 2.11(b) for𝑛 = 2]. The product is shown in Figure 2.11(c), where it is seen that the net area under the product𝑥 (𝑡) 𝑥 (𝑡 + 𝑛Δ)is−Δ, which gives𝑅 (2Δ) = −7ΔΔ = −17for this case.

In fact, this answer results for any𝜏equal to a nonzero integer multiple ofΔ. For𝜏 = 0, the net area under the product𝑥 (𝑡) 𝑥 (𝑡 + 0)is7Δ, which gives𝑅 (0) =7Δ7Δ = 1. These correlation results are shown in Figure 2.11(d) by the open circles where it is noted that they repeat each𝜏 = 7Δ. For a given noninteger delay value, the autocorrelation function is obtained as the linear interpolation of the autocorrelation function values for the integer delays bracketing the desired delay value. One can see that this is the case by considering the integral∫𝑇0𝑥(𝑡)𝑥(𝑡 + 𝜏) 𝑑𝑡and noting that the area under the product𝑥(𝑡)𝑥(𝑡 + 𝜏) must be a linear function of𝜏due to𝑥(𝑡)being composed of square pulses. Thus, the autocorrelation function is as shown in Figure 2.11(d) by the solid line. For one period, it can be expressed as

𝑅 (𝜏) =8 7Λ(𝜏

Δ )− 1

7 , |𝜏|≤ 𝑇0

2

The power spectral density is the Fourier transform of the autocorrelation function, which can be obtained by applying (2.146). The detailed derivation of it is left to the problems. The result is

𝑆 (𝑓) = 8 49

∑∞ 𝑛=−∞

sinc2 ( 𝑛

7Δ )𝛿(

𝑓 − 𝑛7Δ )− 1

7 𝛿(𝑓)

0 1 0 –1

2 4 6 8 10 12 14

t, s

x(t)

0 1 0 –1

2 4 6 8 10 12 14

t, s

x(t – 2Δ)x (t) x(t – 2Δ)

0 1 0

–1 2 4 6 8 10 12 14

t, s

–6 –4 –2

1 0.5 0

2 4

0 6

t, s

R(τ)

–1 0.2 0.1

0 –0.8 –0.6 –0.4 –0.2 0 0.2 0.4 0.6 0.8 1

f , Hz

S(f)

(a)

(b)

(c)

(d)

(e)

Figure 2.11

Waveforms pertinent to computing the autocorrelation function and power spectrum of anm-sequence of length 7.

and is shown in Figure 2.11(e). Note that near𝑓 = 0,𝑆 (𝑓) =(8

49−17)

𝛿 (𝑓) =491𝛿 (𝑓), which says that the DC power is491 = 712 watts. The student should think about why this is the correct result. (Hint:

What is the DC value of𝑥(𝑡)and to what power does this correspond?)

■ The autocorrelation function and power spectral density are important tools for systems analysis involving random signals.