EECS 20 Chapter 5 Part 21 Linear Infinite State Systems Last time we Looked at systems with an infinite number of states (Reals N ) Examined property of linearity Formulated (A, B, C, D) system definition Derived equation to calculate state response Today we will Characterize linear discrete-time systems by their impulse response Formulate zero-input output response using impulse response Compute impulse response for example systems See examples of FIR and IIR systems

EECS 20 Chapter 5 Part 22 Zero-State and Zero-Input Responses Last time, we came up with equations that give us s(n) and y(n): s(n) = A n s(0) + Σ A n-1-m B x(m) y(n) = C A n s(0) + Σ C A n-1-m B x(m) + D x(n) We gave names to the terms in these equations: Zero-input state response:A n s(0) Zero-input output response:C A n s(0) Zero-state state response: Σ A n-1-m B x(m) Zero-state output response: Σ C A n-1-m B x(m) + D x(n) m=0 n-1 m=0 n-1 m=0 n-1 m=0 n-1

EECS 20 Chapter 5 Part 23 Impulse Function The general system response can also be described by its response to a particular input signal called the impulse:  : Integers  {0, 1} "n  Integers,  (n) = This function is also known as the Kronecker Delta function. 1if n = 0 0if n ≠ 0 {

EECS 20 Chapter 5 Part 24 Impulse Response When we apply the impulse function  as the input x to a SISO system with an initial state of zero, we denote the resulting output y by h. This special output, h, is called the impulse response. Using the formula for the zero-state output response, y(n) = Σ C A n-1-m B x(m) + D x(n) We see that when the  function is substituted for x, m=0 n-1 h(0) = Dh(1) = CBh(2) = CABh(3) = CA 2 B h(n) = CA n-1 B for n > 0h(0) = Dh(n) = 0 for n < 0

EECS 20 Chapter 5 Part 25 General Response Using Impulse Response Looking again at the zero-state output response, y(n) = Σ C A n-1-m B x(m) + D x(n) we see that we can write a general output y(n) using the impulse response h: y(n) = Σ h(n-m) x(m) + h(0) x(n) = Σ h(n-m) x(m) m=0 n-1 m=0 n n-1 Since h(n) = 0 for n < 0, y(n) = Σ h(n-m) x(m) m=-∞ ∞

EECS 20 Chapter 5 Part 26 Convolution This operation being performed on h and x is called convolution, and it is denoted by  y(n) = Σ h(n-m) x(m)  y = h  x The convolution operation is commutative: h  x = x  h Σ h(n-m) x(m) = Σ h(k) x(n-k) The convolution operation has useful properties in the frequency domain, covered in later chapters. That is one of the reasons we use the impulse response characterization. m=-∞ ∞ ∞ k=-∞ ∞

EECS 20 Chapter 5 Part 27 Example: FIR Consider the 7-day moving average: y(n) = 1/7 Σ x(n-k) Find the impulse response. h(n) = 1/7 Σ  (n-k) The  (n-k) will equal 1 for k = n, 0 otherwise. We can have k = n for n between 0 and 6. h(n) = 1/7 for 0 ≤ n ≤ 6h(n) = 0 otherwise k=0 6 6 The impulse response has finite duration (h(n) = 0 for all n > 6). If a system has an impulse response with finite duration, we call it a finite impulse response (FIR) system.

EECS 20 Chapter 5 Part 28 Example: IIR Consider the oscillator system, and find the impulse response: cos (  ) -sin (  ) sin (  ) cos (  ) We showed that We can also show that cos (n  ) -sin (n  ) sin (n  ) cos (n  ) A = C =1 0B = 0101 D = 0 h(n) = CA n-1 B for n > 0h(0) = Dh(n) = 0 for n < 0 A n = The impulse response does not die out; there is no N for which h(n) = 0 for n>N. We say these systems have infinite impulse response (IIR). h(n) = -sin( (n-1)  ) for n>0 h(n) = 0 otherwise

EECS 20 Chapter 5 Part 29 Example: Unstable Consider the bank interest system s(n+1) = (1+r)s(n) + x(n) y(n) = s(n) Find the impulse response h. Here, the impulse response has an interesting interpretation: it is the yearly value of a bank account when $1 is deposited in year 0, and nothing is deposited thereafter. We may show that h(n) = (1+r) n Here, the impulse response does not die out; it is an IIR system. In addition, h(n) increases with n, without bound. If a system produces an unbounded output for a bounded input, we say the system is unstable.

EECS 20 Chapter 5 Part 210 MIMO Systems Since the impulse function  takes values in Reals, the impulse response h is technically defined only for single-input systems. We may define a matrix-valued function h with the same values and the general output y will be given by the same convolution y(n) = Σ h(n-m) x(m)  y = h  x So we can still have an h with the same properties for general MIMO systems, but we won’t call it an impulse response. h(n) = CA n-1 B for n > 0h(0) = Dh(n) = 0 for n < 0 m=-∞ ∞