Correlated Random Processes Correlated random processes can be described by 1 The autocorrelation function 2 The power spectrum Random Processes IV 3 A lter with a white noise input Lecture 14 Spring 2002 The three descriptions are related Each provides a di erent perspective on the random process Lecture 14 Generating a Correlated Random Process 1 Di erence Equation Digital Filter Model We will provide a means to generate correlated random processes If you can provide a generator you have certainly provided one description of the rp Later we will look at how to determine the parameters of a lter model that generates a given rp A digital lter can be described in several ways Di erence equation Block diagram Impulse response We will only work here with WSS random processes System function The descriptions are equivalent but each provides a di erent insight These descriptions are related to those of a random process when the lter input is white noise Lecture 14 2 Lecture 14 3 Di erence Equation Model Di erence Equation Model Let the input sequence to a digital lter be denoted by x n and the corresponding response by y n The current output value can depend upon the current input value past input values and past output values The di erence equation can be solved for yn in terms of the current and past inputs and past outputs a1y n 1 a2y n 2 apy n p The properties of the output sequence are determined by y n a1y n 1 a2y n 2 apy n p b0x n b1x n 1 bq x n q The properties of the input sequence The di erence equation is completely described by the coe cients 1 a1 ap and b0 b1 bp We will always assume without loss of generality that a0 1 Lecture 14 y n b0x n b1x n 1 bq x n q 4 The lter coe cients Lecture 14 5 Block Diagram Example Single pole Filter Let x n 0 1 0 0 0 0 and choose the origin so that x 0 1 y n x n ay n 1 Assume initial y 1 0 n x n y n Lecture 14 6 Lecture 14 0 1 1 1 0 a 2 0 a2 condition 3 0 a3 4 0 a4 7 Impulse Response Impulse Response The impulse response of a digital lter is the sequence that is generated when the input sequence is x n n where If the impulse response of a discrete system is h n then the response to any input sequence x n can be computed by the convolution n 0 n 0 1 n 0 y n m x m h n m The impulse response of the single pole lter is y n h n a n for n 0 For the previous example The output is clearly bounded if and only if a 1 y n We will see later when we discuss the system function that this system has a single pole whose location is determined by a and which is stable if the pole is inside the unit circle Lecture 14 8 m x m a n mstep n m a k x n k k 0 This provides an exponential weighting of the past inputs Lecture 14 9 System Function Model System Function The system function H z describes the response of the system to an exponential input sequence x n z n z may be any complex number often expressed in the form z rei Assume that the response is of the form y n H z z n Note that H z y 0 is a number once z has been speci ed Substitute both sequences into the di erence equation and eliminate common z n terms H z b0 b1z 1 bq z q 1 a1z 1 a2z 2 apz p The close relationship between the system function the di erence equation and the block diagram is evident The system function is de ned wherever the denominator is not zero 1 a1z 1 a2z 2 apz p z nH z b0 b1z 1 bq z q z n The roots of the denominator that are not cancelled by roots of the numerator are called the poles of the system function Lecture 14 Lecture 14 10 11 Z Transform System Models The z transform is de ned for discrete sequences It is closely related to the Fourier transform for a discrete sequence and contains information about the frequencies contained in the sequence The de nition is simply X z We will illustrate the computation of the z transform for some example functions b0 b1z 1 bq z q 1 a1z 1 a2z 2 apz p Lecture 14 n x n z n The inverse z transform can be de ned but is not necessary in this analysis y n b0x n b1x n 1 bq x n q a1y n 1 a2y n 2 apy n p H z 12 Lecture 14 13 Example Exponential Sequence Delay Operator x n Anstep n Suppose that the z transform of x n is X z Then the z transform of x n k is X z z k Let where A is a constant The sequence is decreasing in magnitude if A 1 X z n 0 Anz n X z z k A z n n n n 0 This is just a geometric series which converges provided A z 1 Hence 1 z X z 1 A z z A x n z n k x m k z m We can therefore refer to z 1 as the delay operator The denominator has a root pole at z A The sequence converges if the pole is inside the unit circle in the z plane Lecture 14 14 Lecture 14 15 Di erence Equation and System Function Di erence Equation and System Function Take the z transform of all terms in the di erence equation by making use of the delay operator Because all operations are linear Y z a1Y z z 1 a2Y z z 2 apY z z p b0X z b1X z z 1 bq X z z q It is natural to de ne the polynomials A z 1 a1z 1 a2z 2 apz p B z b0 b1z 1 bq z q This can be rearranged as a ratio of polynomials in z that are identical to the system equation b0 b1z 1 bq z q Y z H z X z 1 a1z 1 a2z 2 apz p These are the z transforms of the denominator and numerator polynomials in the system function and also polynomials formed from the di erence equation coe cients The preceding results then show the following useful relationships H z This leads to the system function equation A z Y z B z X z Y z H z X z Lecture 14 16 System Function and Impulse Response y n m Y z n x m h n m H z h k z k k For the single pole lter y n z n 17 The …