On a second reading if you are interested you may read some or all of the footnotes If you are even more i nterested you can come to o64259ce hours Another resource is Appendix A of the course text book 1 1 The Two Main Properties 11 The Power Densi ID: 29145 Download Pdf

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On a second reading if you are interested you may read some or all of the footnotes If you are even more i nterested you can come to o64259ce hours Another resource is Appendix A of the course text book 1 1 The Two Main Properties 11 The Power Densi

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EECS 123 Digital Signal Processing University of Californi a, Berkeley: Fall 2007 Gastpar November 30, 2007 The Power Density Spectrum Preliminary version — please report errors, typos, and sugg estions for improvements This handout is designed to be read without any of the footnotes. On a second reading, if you are interested, you may read some or all of the footnotes. If you are even more i nterested, you can come to oﬃce hours. Another resource is Appendix A of the course text book [1]. 1 The Two Main Properties 1.1 The Power Density Spectrum We start by remembering

that the power of a signal ] is deﬁned as = lim + 1 (1) While this formula may look a little complicated, it is reall y just the average of the squares of the signal values! (You have seen this formula in EECS 120.) A random signal ] can be characterized by its power density spectrum xx (2) which is a 2 -periodic function saying, typically, how much power the si gnal ] has at frequency ω, for π < π. Because power is never negative, we must have that xx An example of a power spectrum for a signal ] is given in Figure 1. xx π/2 −π/2 P ( ) Figure 1: An

example of a power spectrum: This signal has “mor e power at low frequencies. Property 1. The power of a random signal can be found from its power density spectrum xx simply by integrating: xx dω. (3) More precisely, a class of random signals called stationary and ergodic can be characterized in this fashion.

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Equivalently, the power of a random signal can also be found as in Equation (1). For the signal ] whose power spectrum is given in Figure 1, we ﬁnd xx d (4) π/ d π/ π/ d π/ d (5) + 3 (6) = 2 (7) 1.2 White Noise A very important random

signal is called white noise, for us denoted by Its special property is that the power spectrum is ﬂat: ee ) = for all ω. (8) It is easy to verify that the power of the white noise ] is then simply ee d (9) 1.3 The Filtering Property The most important (and useful) property of the power densit y spectrum concerns LTI systems (i.e., ﬁlters): x[n] y[n] H(z) Figure 2: Passing a signal ] through an LTI system as we have done many times. Now, is a random signal. Property 2. If we pass a random signal through an LTI system with transfer function the the output is also a random

signal, call it and its power density spectrum is given by yy ) = j xx (10) Let’s again look at the signal ] with power density spectrum as given in Figure 1. An example of ﬁltering this signal is shown in Figure 3. This is a bit surprising: since the signal is random, really, the power as in Equation (1) is also random. However, for the important class of ergodic signals, the limit converges to a constant.

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H(e ) −π/2 −π/4 −3π/4 −π π/4 3π/4 π/2 −π/2 −π/4 −3π/4 −π

π/4 3π/4 π/2 −π/2 P ( ) xx P ( ) yy π/2 Figure 3: The signal ] is passed through the ﬁlter j resulting in a signal ] with diﬀerent power density spectrum.

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2 Downsampling and Upsampling 2.1 Downsampling x[n] w[n] Figure 4: Downsampling by a factor of M. Property 3. Downsampling does not change the power: (11) Property 4. Let be white noise. Then, is also white noise. Note: If the input ] is not white noise, then it is a bit more tricky to ﬁnd the power densi ty spectrum of This is beyond the scope of this class. 2.2 Upsampling

x[n] v[n] Figure 5: Upsampling by a factor of M. Property 5. Upsampling reduces the power by a factor of (12) Property 6. Let be white noise. Then, is also white noise. Note: If the input ] is not white noise, then it is a bit more tricky to ﬁnd the power densi ty spectrum of This is beyond the scope of this class. References [1] A. Oppenheim, R. Schafer, and J. Buck, Discrete-time Signal Processing . Upper Saddle River, NJ: Prentice Hall, 2nd ed., 1999.

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