EECS  Digital Signal Processing University of Californi a Berkeley Fall  Gastpar November   The Power Density Spectrum Preliminary version  please report errors typos and sugg estions for improvement
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EECS Digital Signal Processing University of Californi a Berkeley Fall Gastpar November The Power Density Spectrum Preliminary version please report errors typos and sugg estions for improvement

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 Digital Signal Processing University of Californi a Berkeley Fall Gastpar November The Power Density Spectrum Preliminary version please report errors typos and sugg estions for improvement




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Presentation on theme: "EECS Digital Signal Processing University of Californi a Berkeley Fall Gastpar November The Power Density Spectrum Preliminary version please report errors typos and sugg estions for improvement"— Presentation transcript:


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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) Lets 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.