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Unformatted text preview: Recursive and Nonrecursive Discrete-Time Systems If a system output y(n) at time n depends on any number of past output value y(n-1), y(n-2),... , it is called a recursive system. Let's have a DTS that gives the cumulative average y ( n) = 1 n x(k ) n + 1 k =0
n -1 k =0 We can find y(n) more efficient by utilizing the output y(n-1).
y (n ) = = 1 n +1 x(k ) + x(n ) 1 ( ny (n - 1) + x(n ) ) n +1 n 1 = y ( n - 1) + x(n ) n +1 n +1 1 Recursive and Nonrecursive Discrete-Time Systems
+ X y(n) 1 n+1
X z-1 n This system requires two multiplication, one addition, and one memory location. This is a recursive system which means the output at time n depends on any number of a past output values. So, a recursive system has feed back output of the system into the input. This feed back loop contains a delay element. y (0) = x (0)
y (1) = 1 1 y (0) + x(1) 2 2 y (2) =
y ( n0 ) = 2 1 y (1) + x(2) 3 3
n0 1 y ( n0 - 1) + x( n0 ) n0 + 1 n0 + 1 Recursive and Nonrecursive Discrete-Time Systems For the causal FIR systems
y ( n) =
M k =0 h( k ) x ( n - k ) = F [ x(n), x(n - 1),..., x( n - M ) ] = h(0) x(n) + h(1) x(n - 1) + ... + h( M ) x(n - M ) x(n) F[x(n),x(n-1), ...,x(n-M)] y(n) x(n) F[y(n-1),..y(n-N), x(n),...,x(n-M)] z-1 y(n) Nonrecursive system Recursive system 2 !
" $ # ' % ( & & & Let's have a recursive system that is first-order difference
X(n) y(n) + a y ( n) = ay ( n - 1) + x ( n)
z-1 where is a constant and system is time invariant. We assume that we have initial condition y(-1). For , y(n) can be obtained
y (0) = ay (-1) + x(0)
y (1) = ay (0) + x (1) = a 2 y (-1) + ax (0) + x(1)
n k =0 y (n) = a n+1 y ( -1) + a n x(0) + a n-1 x(1) + ... + x( n) y (n) = a n+1 y (-1) + a k x(n - k )
The response y(n) of the system depends on initial condition y(-1) of the system and the system response to the input signal If the system is initially relaxed at time n=0, its memory should be zero. So, y(-1)=0. Then, system is at zero state and the corresponding output is called zero-state response or forced response.
y zs (n) =
n k =0 a k x(n - k ) n0 If system is initially nonrelaxed (y(-1)0) and the input x(n)=0 for all n. The corresponding output is called zero-input response or natural response. y zi (n) = a n+1 y (-1) n0 3 !
The total system response is y (n) = yzs (n) + y zi (n)
General form of linear constant-coefficient difference equation is
y ( n) = -
N k =0
N k =1 M k =0 ak y ( n - k ) +
M k =0 bk x (n - k ) ak y ( n - k ) = bk x (n - k ) In order to find y(n), we need to know initial conditions y(n-1), y(-2),...,y(n-N) and the input x(n) for all n>=0. Recursive system may be relaxed or non-relaxed, depending on the initial condition. !
A system is linear if satisfy the following requirements: 1. The total response is equal to the sum of the zero-input and zero-state responses y (n) = y zs (n) + yzi (n)
2. The principles of superposition applies to the zero-state response (zero-state linear) 3. The principles of superposition applies to the zero-input response (zero-input linear) If a system does not satisfy all three separate requirement, system is called nonlinear. ) 4 ...
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- Winter '08