Lecture 02 - LTI description

# Lecture 02 - LTI description - 1 Lecture 2 Systems...

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1 Lecture 2: System’s description (LTI, causal, finite dimensional) Instructor: Dr. Gleb V. Tcheslavski Contact: [email protected] Office Hours: TR 11:00-12:00; Room 2030 Class web site: http://ee.lamar.edu/gleb/dsp/ind ex.htm ELEN 5346/4304 DSP and Filter Design Fall 2008 2 The Signal Flow Graph - nk m nm m Let a y b x ΝΜ κ− κ=0 =0 = ∑∑ for a BIBO stable LTI system 0 1 nn k m n m m Assume a y a y b x κ −− κ=1 = + y n is a function of previous inputs and outputs, therefore the system is causal. If a k and b m are constants Linear Constant coefficient Difference Equation * Δ y n x n b 0 (2.2.1) ELEN 5346/4304 DSP and Filter Design Fall 2008 + + + + memory * Δ b 1 b 2 z -1 z -1 z -1 z -1 z -1 z -1 a 1 a 2 y n-1 y n-2 x n-1 x n-2 Signal Flow Graph (SFG): Remark: because we discretize the signal by amplitude, the system becomes non-linear!

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3 Difference equation nn k m n m m ya yb x ΝΜ κ −− κ=1 =0 =− + ∑∑ (2.3.1) y n is a function of a k , b m , x n , memory = states. 0 nk n k n k or y h x h x h x κ≥0 κ>0 = = + If we know the initial conditions, we can solve the difference equation: 1, 1, 2, ' xy predictions of system s behavior (2.3.2) ELEN 5346/4304 DSP and Filter Design Fall 2008 If (2.3.1) is a linear constant coefficient difference equation, the output consists of a homogeneous solution and a particular solution: y n = y h,n + y p,n (2.3.3) 4 Homogeneous (complementary) solution 0, 0 n x n the output is due to the initial conditions Homogeneous equation: 0 N ay = (241 0 kn k k = 0 '0 N k k Let s assume y a λλ = = = 1 01 1 ( ) 0 is a characteristic equation nN N N Then a a a a λ + + + + = The characteristic equation has N characteristic roots: λ 1 , … λ N (2.4.1) (2.4.2) (2.4.3) ELEN 5346/4304 DSP and Filter Design Fall 2008 • If the system is real, x n , y n , and all the coefficients (a, b) are real • For LTI systems h n are real
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Lecture 02 - LTI description - 1 Lecture 2 Systems...

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