1 / 18

Lecture 3: Cascaded LTI systems, SFGs, and Stability…

Lecture 3: Cascaded LTI systems, SFGs, and Stability…. Instructor: Dr. Gleb V. Tcheslavski Contact: gleb@ee.lamar.edu Office Hours: Room 2030 Class web site: http://ee.lamar.edu/gleb/dsp/index.htm. Cascaded LTI. y n. y n. v n. w n. x n. x n. (a). h 1,n.

kendall
Download Presentation

Lecture 3: Cascaded LTI systems, SFGs, and Stability…

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Lecture 3: Cascaded LTI systems, SFGs, and Stability… Instructor: Dr. Gleb V. Tcheslavski Contact:gleb@ee.lamar.edu Office Hours:Room 2030 Class web site:http://ee.lamar.edu/gleb/dsp/index.htm

  2. Cascaded LTI yn yn vn wn xn xn (a) h1,n h2,n h1,n h2,n (3.2.1) (3.2.2) According to (3.2.2), system (a) is equivalent to (b): (b) Note: this property is a result of linearity

  3. + + + + + + + + + + + + Fundamental direct forms of Signal Flow Graph (SFG) vn yn S3n S1n Fundamental Direct form SFG z-1 -a1 S2n S4n z-1 -a2 xn b0 h1,n h2,n z-1 … z-1 b1 xn wn yn xn wn yn z-1 b0 b0 b2 z-1 z-1 z-1 z-1 -a1 b1 b1 -a1 … wn-1 z-1 z-1 z-1 b2 b2 -a2 -a2 wn-2 z-1 z-1 … … z-1 … … h1,n h2,n Equivalent (Direct 2) form

  4. SFG description for the Fundamental Direct form SFG (for SOS): Next state: (3.4.1) Output: (3.4.2) (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: (3.5.1) (3.5.3) (3.5.2) (3.5.4) (3.5.5) (3.5.6)

  6. SFG description (cont 2) (3.6.1) (3.6.2) Holds only for n-1-l ≥ 0 (3.6.3) For the DF 2, the initial state depends on initial conditions but it’s NOT the same! Fundamental Direct form: {A, b, c, d} Direct 2 form: These systems are equivalent in terms of input/output!

  7. SFG description (cont 3) For the DF2 (3.5.3): Q.: can we find S-3? It’s not trivial since we would need to know x-2 etc. However, we don’t really need it!

  8. SFG relations Let’s say we need to find if is known… (3.8.1) (3.8.2) fundamental direct 2 (3.8.3) In our case, we only need two equations (n=0 and n=1) (3.8.4)

  9. SFG and characteristic roots (3.9.1) (3.9.2)  - eigenvalue (characteristic root) (3.9.3) (3.9.4) (3.9.1)  (3.9.4) !!! (3.9.5)

  10. Stability triangle for a causal BIBO system: (3.10.1) (3.10.2) (3.10.3) (3.10.4) (3.10.5) Stability triangle (3.10.6) Coefficients must be inside the triangle For systems of higher orders, there are no simple conditions – can always cascade!

  11. Stability triangle The parabola a2 = a12/4 splits the stability triangle into two regions. The region below it corresponds to real and distinct roots 1, 2: Impulse response: (3.11.1) The points on the parabola result in real and equal (double) roots (poles) 1, 2: Impulse response: (3.11.2)

  12. Stability triangle The points above the parabola (a12 < 4a2) correspond to complex-conjugate roots 1, 2: Assuming the roots (poles) in the form: (3.12.1) the filter coefficients are: (3.12.2) And the corresponding impulse response: (3.12.3) has a decaying oscillatory behavior.

  13. More on SFG relations (3.13.1) (3.13.2) (3.13.3) (3.13.4) (3.13.5) Many systems are equivalent in terms of I/O, hn but are different inside! State space description is not unique in terms “what’s going inside”!

  14. + + + + Overflow control (3.14.1) Fixed-point representation may cause a state overflow! xn cs cs-1 b0 yn z-1 z-1 -a1 b1 Consider a second order system z-1 S1,n yn/ z-1 -a2 b2 S3,n S2,n S4,n (3.14.2) We cannot change the system - it’s given. Instead, we add a scaling coefficient csand cs-1 to compensate for it.

  15. Overflow control (cont) Determinant of SFG: We may use the original hn to find the output  find cs – experiment! cs does not change the determinant of SFG.

  16. + + + + Overflow control (cont 2) If we have a cascade… To make sure that the input of the 2nd cascade is bounded cs-1 xn cs cs(2) xn cs-1 csb0 yn We can also embed cs into b-coeffs z-1 z-1 -a1/2 csb1 2 z-1 z-1 -a2/2 csb2 Implementations: a2 1 a1 2 1 1 This way we add quantization noise! -1

  17. + + + + + Overflow control (cont 3) Alternative implementations: Drawbacks? 1st option: cs-1 xn 1/2 2 Add additional components to the system 2nd option: xn cs-1 csb0 yn z-1 z-1 -a1/2 csb1 z-1 -a1/2 csb2 z-1 It’s better to NOT divide a2 since it is less than 1  this way we don’t introduce quantization noise. We chose to divide by 2 since 2 is a minimum divider that bounds a1 to 1. -a2

  18. Overflow… Assume we use 8 bits to represent integer numbers… obviously, from 0 to 255. If A = 255 and B is just 1… C = A + B = 256… which cannot be represented by 8 bits. As a result back

More Related