Difference Equations

Transcription

1 Difference Equations Andrew W H House 10 June The Basics of Difference Equations Recall that in a previous section we saw that IIR systems cannot be evaluated using the convolution sum because it would require an infinite number of operations We showed, however, that sometimes difference equations can be used to evaluate/implement such systems Thus, an important class of DT LTI systems are those whose inputs/outputs satisfy linear constant-coefficient (LCC) difference equations of the following form Note that this is the DT version of the LCC differential equation in CT y(n) + a 1 y(n 1) + a y(n ) + + a N y(n N) = b 0 x(n) + b 1 x(n 1) + b x(n ) + + b N x(n N) To solve this general difference equation, we need N initial conditions Now, if we wish to solve the above equation causally, for example, we must rewrite it to focus on the current output, y(n) y(n) = b 0 x(n) + b 1 x(n 1) + b x(n ) + + b N x(n N) a 1 y(n 1) a y(n ) a N y(n N) Suppose x(n) = u(n), so we start calculating y(n) at n = 1,, 3, To do this, we need N initial conditions: y( 1), y( ), y( N) If the equation under consideration is to represent a strictly LTI system, the initial conditions (IC s) must be zero [y( 1) = y( ) = = y( N) = 0] and must float with the input ENGI 784 Discrete-Time Systems and Signals 1

2 This last condition means that if, say, x(n) = u(n 1), then we start y(n) at n = 1, 13, 14, so we need [y(11) = y(10) = = y(1 N) = 0] We can encapsulate this requirement with the initial rest condition Definition of IR (initial rest): If x(n) = 0 for n < n o, then y(n) = 0 for n < n o (Note that IR only describes causal systems) 11 Whence Difference Equations Where do difference equations come from? There are a number of possible sources 1 Difference equations can be exact models of DT systems Example 11: Interest-Bearing Bank Account y(n) = 1αy(n 1) + x(n) A bank account could be considered a naturally discrete system In the above equation, y(n) is today s balance, y(n 1) is yesterday s balance, α is the interest rate, and x(n) is the current day s net deposit/withdrawal Difference equations can be approximations of CT differential equations We can sample signals in a CT differential equation, as shown in the following example Example 1: Sampling a CT Differential Equation Consider a CT differential equation, y (t) + y (t) + y(t) = x(t) x (t), IR and suppose we sample, x d (n) = x(t) t=nt, y d (n) = y(t) t=nt, T > 0 then what difference equation can process these signals? Recall in CT, and so define y y(t) y(t t) (t) = lim t 0 t y d(n) = y (t) t=nt ENGI 784 Discrete-Time Systems and Signals

3 which leads to the following approximation of y d (n) y d(n) = y d(n) y c (n 1) T Similarly, we can find y d (n) using y d (n) as the basis y d(n) = y d (n) y c(n 1) T We can likewise also find x (n) from x(t) and x (t) Substituting the DT values into our CT equation, we get the following overall difference equation for the DT system ( ) y d (n) y d (n 1) + y d (n ) yd (n) y d (n 1) + T T = x d (n) x d(n) x d (n 1) T + y d (n), IR We can group the like terms to get a more proper-looking difference equation ( 1 T y d(n ) T + ) ( 1 y T d (n 1) + T + ) T + 1 y d (n) = ( T ) x d (n) + 1 T x d(n 1), IR 3 Difference equations can be designed to produce desired effects in DT In other words, the difference equation can be used to implement the system 1 Sample Difference Equations Let s look at some sample difference equations 1 y(n) = x(n 5) is a pure delay system Here, = 1, b 5 = 1, and all other values a k, b k = 0 y(n) = x(n) x(n 1) is a differencer In this case, = b 0 = b 1 = 1 and all other values a k, b k = 0 Both of these cases are non-recursive difference equations, since the only appearing value of the output is the current y(n) ENGI 784 Discrete-Time Systems and Signals 3

4 3 The system y(n) = x(k) is an accumulator We can rewrite this as a difference equation that is recursive on y(n) n 1 y(n) = x(k) = x(n) + x(k) = x(n) + y(n 1) So y(n) y(n 1) = x(n), IR As we said, this is clearly recursive on y(n) 4 We have seen the system y(n) = x(k) ( ) n k 1 = x(n) and shown that it can be written as a difference equation ( ) n k 1 y(n) = x(k) n 1 ( ) n k 1 = x(n) + x(k) = x(n) + 1 n 1 ( 1 x(k) = x(n) + 1 y(n 1) ( ) n 1 u(n) ) n k 1 So y(n) 1 y(n 1) = x(n), IR This is a recursive difference equation that is equivalent to the convolution sum Using Difference Equations What can we do with difference equations? 1 We can implement DT LTI systems (This is illustrated in the handout and in the next section) We can get the difference equation from the h(n) of a DT LTI system, so the difference equation is equivalent the convolution sum and can be used to evaluate convolution For example, if ( ) n 1 h(n) = u(n) ENGI 784 Discrete-Time Systems and Signals 4

5 then we can write it as a difference equation so x(n) h(n) y(n) = x(k) y(n) = x(n) + 1 y(n 1), ( ) n k 1 = x(n) + 1 y(n 1), IR IR and we can implement the difference equation 3 We can solve difference equations for a given x(n), either in time (recursively or via formal methods like the CT differential equation) or in frequency (which we ll cover later) The recursive solution is an actual system implementation Example 1: Solution of Difference Equations Consider the following system y(n) 1 y(n 1) = x(n), IR We want to find y(n) if x(n) = u(n) The first step is to write the equation causally y(n) = x(n) + 1 y(n 1), IR The IR condition means that y(n) = 0 for n < 0 So, we can evaluate recursively from n = 0 y(0) = x(0) + 1 y(0 1) = = 1 y(1) = x(1) + 1y(1 1) = = 3 y() = x() + 1y( 1) = = 7 4 y(3) = x(3) + 1y(3 1) = = 15 8 We notice a trend in the values, so we can express the output as follows [ ( ) n 1 y(n) = u(n)] u(n) = s(n) This is the step response ENGI 784 Discrete-Time Systems and Signals 5

6 Let s try this for x(n) = δ(n) Again, evaluating recursively from n = 0 y(0) = x(0) + 1 y(0 1) = = 1 y(1) = x(1) + 1y(1 1) = = 1 y() = x() + 1y( 1) = = 1 4 y(3) = x(3) + 1y(3 1) = = 1 8 Not surprisingly, we see a trend here, and as expected, this output is the impulse response of the system ( ) n 1 y(n) = h(n) = u(n) This means of solution will work for any input x(n) Consider the following x(n) We see that x(n) = 0 for n < 1, so y(n) = 0 for n < 1 due to IR condition We evaluate recursively from n = 1 y(1) = x(1) + 1 y(1 1) = = 1 y() = x() + 1y( 1) = = 5 y(3) = x(3) + 1y(3 1) = = 9 4 y(4) = x(4) + 1y(4 1) = = 1 8 y(5) = x(5) + 1y(5 1) = = y(6) = x(6) + 1y(6 1) = = 17 3 y(7) = x(7) + 1y(7 1) = = ENGI 784 Discrete-Time Systems and Signals 6

7 There is no overall form to represent this output, but we can see that y(n) for n > 7 are going to follow the trend of 17/ n 1 that was apparent since n = 5 We can plot y(n), shown below So this recursive method of solutions works for any x(n) so long as we have IR, and is easily implementable in software or hardware 3 Graphical Representation of Difference Equations Note: This material summarizes the handout We can represent the general form of LCC difference equations as where a k, b k are scalars N a k y(n k) = k=0 N b k x(n k) k=0 Clearly, there are only three operations involved: multiplication by a scalar, addition, and shifting in time Shifting in time is done via the unit delay component We can represent these graphically, shown below ENGI 784 Discrete-Time Systems and Signals 7

8 That means we have an easy way to visualize difference equations Furthermore, this graphical representation lends itself to analysis Say we had the difference equation y(n) = x(n) + x(n 1) 3y(n 1), IR which is represented graphically as shown below (Note that we can group the input terms and can represent them as w(n)) We can swap the two halves of the graphical representation, and then we can combine the unit delays to reduce the storage requirements of the implementation This, this form of graphical analysis provides tools and techniques allowing more efficient implementation ENGI 784 Discrete-Time Systems and Signals 8

9 There is some question as to why this swapping of halves of the difference equation implementation can work We can consider this in the context of the above sample difference equation We already stated that y(n) = w(n) 3y(n 1), IR where w(n) = x(n) + x(n 1) for our initial graphical representation For our swapped and reduced version, we have a different intermediate signal, z(n), and we see the following relationships z(n) = x(n) 3z(n 1) y(n) = z(n) + z(n 1) This y(n) does not look much like our originally defined y(n) However, if we substitute the value of z(n) into this new equation for y(n) for both z(n) and z(n 1) we see the following y(n) = z(n) + z(n 1) = [x(n) 3z(n 1)] + [x(n 1) 3z(n )] = x(n) 3z(n 1) + x(n 1) 6z(n ) = x(n) + x(n 1) 3z(n 1) 6z(n ) = x(n) + x(n 1) 3 [z(n 1) + z(n )] = x(n) + x(n 1) 3y(n 1) if we consider y(n) in terms of z(n) Thus, the new form of the difference equation is still equivalent to the original, but has allowed us to remove one of the unit delay components and thus provide a more efficient implementation This sort of optimization would be difficult to find without using these graphical techniques Signals and Systems, E: Chapter 4 43, page(s) Signal Processing First: Chapter 5-, page(s) 103 ENGI 784 Discrete-Time Systems and Signals 9