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# Lecture 03 - Cascaded LTI - Lecture 3 Cascaded LTI systems...

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1 Lecture 3: Cascaded LTI systems, SFGs, and Stability… Instructor: Dr. Gleb V. Tcheslavski Contact: [email protected] Office Hours: Room 2030 Class web site: http://ee.lamar.edu/gleb/dsp/ind ex.htm ELEN 5346/4304 DSP and Filter Design Fall 2008 2 Cascaded LTI h 1,n h 2,n x n v n y n (a) 2, 1, 1, ( ) 1 nn k k l n k l l k n l k n n ll k yh v h h x h h x h w 2,κ − − − − κκ == = = ∑∑ 1, 1, n n n n n wh x yhh x h x = = = (3.2.1) (3.2.2) According to (3.2.2), system (a) is equivalent to (b): ELEN 5346/4304 DSP and Filter Design Fall 2008 (b) h 2,n h 1,n x n w n y n Note: this property is a result of linearity

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3 Fundamental direct forms of Signal Flow Graph (SFG) + x n b 0 + y n v n nn k m n m m ya yb x ΝΜ κ −− κ=1 =0 =− + ∑∑ + b 1 b 2 z -1 z -1 z -1 + z -1 z -1 z -1 -a 1 -a 2 Fundamental Direct form SFG S 1n S 2n S 3n S 4n h 1,n h 2,n + x n b 0 y n w n y n + b 0 w n x n κ1 0 ELEN 5346/4304 DSP and Filter Design Fall 2008 + b 1 b 2 z -1 z -1 z -1 + + z -1 z -1 z -1 -a 1 -a 2 h 2,n h 1,n w n-1 w n-2 + b 1 b 2 z -1 z -1 z -1 + + -a 1 -a 2 Equivalent (Direct 2) form 4 SFG description 11 1 00 0 0 S ⎡⎤ for the Fundamental Direct form SFG (for SOS): Next state : 1, 1 2, 1 1 0 3, 1 1 2 1 2 4, 1 0 10 0 0 00 1 0 0 n n n n n S SS x b Sb b a a S + + + + + ⎢⎥ == + ⎣⎦ [ ] 12 1 2 0 n b aa S b x + Output : (3.4.1) (3.4.2) ELEN 5346/4304 DSP and Filter Design Fall 2008 1 n T n SA S b x yc Sd x + =+ (3.4.2) Here {A,b,c,d} are state-space description. They represent one clock-cycle for a piece of soft/hard-ware.
5 SFG description (cont) A – the system matrix b – input matrix c – output matrix d – transmission matrix For the equivalent (Direct 2) form of SFG: 12 1 1 10 0 nn n aa SS x y bb abb aSb x + −− ⎡⎤ =+ ⎢⎥ ⎣⎦ = + (3.5.1) (352 1 n T n SA S b x yc Sd x + (3.5.3) ELEN 5346/4304 DSP and Filter Design Fall 2008 [ ] 1 20 2 0 n (3.5.2) () 0 0 0 0 2 200 1 1 0 0 1 32 2 30 01 2 2 00 1 2 ;; T T T Sb x x S A S Abx bx y c AS bx dx S A S A bx Abx bx y c A S Abx bx dx = + = + = + + = + + = + + + = + + + (3.5.4) (3.5.5) (3.5.6) 6 SFG description (cont 2) 1 1 0 0 n l nl l S Ab x = (3.6.1) {} , , 11 1 1 0 1 0 0; zi n zs n Tn n l T n T n l n l ll y y n l T n l n l l n l l Markov parameters y c AS A bx dx c A hS x cA b dc A b ud δ δδ == = ⎛⎞ + = + ⎜⎟ ⎝⎠ += + ∑∑ ±²³ ±´´² ´ ´³ ±´´²´´³ Holds only for n-1-l 0 (3.6.3) (3.6.2) ELEN 5346/4304 DSP and Filter Design Fall 2008 For the DF 2, the initial state depends on initial conditions but it’s NOT the same!

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