MM3FC Mathematical Modeling 3 LECTURE 5

1. MM3FC Mathematical Modeling 3LECTURE 5 Times Weeks 7,8 & 9. Lectures : Mon,Tues,Wed 10-11am, Rm.1439 Tutorials : Thurs, 10am, Rm. ULT. Clinics : Fri, 8am, Rm.4.503 Dr. Charles Unsworth, Department of Engineering Science, Rm. 4.611 Tel : 373-7599 ext. 82461 Email : c.unsworth@auckland.ac.nz

2. This LectureWhat are we going to cover & Why ? • Frequency Response of Simple Systems. (1st Order Difference System = ‘High Pass System’ ) (2nd Order Difference System = ‘Low Pass System’ ) (Cascaded Systems) (L-point Running Average Filter) • The Dirichlet Function. (needed to understand the frequency response of the L-point Running Average Filter)

3. First Difference System • The first difference system is : y[n] = x[n] – x[n-1] • Has coefficients bk={1,-1} with frequency response : • Thus the magnitude response is : … (5.1)

4. The phase response is :

5. From the magnitude plot • The system completely removes the DC component at w = 0. • However, the high frequencies up towards  are preserved. • Thus, this filter is known as a ‘high pass’ filter. • From the phase plot • We can see linear phase over the preserved frequencies. • For both plots we can see only the frequency range 0 < w <  need to be plotted. • And Magnitude is an EVEN function. • And Phase is an ODD function.

6. The Simple Low-Pass FIR Filter • Believe it or not ! We did this earlier. The difference equation : • y[n] = x[n] + 2x[n-1] +x[n-2] • Gave the frequency response of Example 1, Lecture 4: • The Magnitude plot shows the DC and low frequencies are preserved. • And the high frequencies are removed.

7. H1[w]ejwn y1[n]= H1[w]H2[w]ejwn LTI 2 H2[w] x[n] = ejwn LTI 1 H1[w] LTI 1 H1[n] y2[n]= H2[w]H1[w]ejwn = H1[w]H2[w]ejwn x[n] = ejwn H2[w]ejwn LTI 2 H2[w] y[n] = H[w]ejwn x[n] = ejwn LTI Equivalent H[w] Frequency Response for Cascaded Systems • When 2 LTI systems are in cascade then we ‘convolve’ the individual impulse responses of each system together. • The frequency response of 2 LTI systems in cascade is simply the ‘product’ of the individual frequency responses.

8. Thus, • Example 1 : Two LTI systems have coefficients ak={1,-2} and bk={0,1,1}. Determine their cascaded frequency response, impulse response, difference equation and the co-efficients of an equivalent filter. • H1(w) = 1 – 2e-jw and H2(w) = e-jw + e-2jw • H(w) = H1(w)H2(w) = (1 – 2e-jw)(e-jw + e-2jw) = e-jw + e-2jw – 2e-2jw - 2e-3jw • = e-jw - e-2jw - 2e-3jw • Thus the cascaded impulse response is : h[n] = [n-1] – [n-2] –2[n-3] • Thus, the cascaded difference equation is : y[n] = x[n-1] – x[n-2] –2x[n-3] • The equivalent filter has co-efficients : ck = {0,1,-1,-2} • ( Quite handy if you have 3 or more cascaded systems) … (5.2)

9. Frequency Response of an L-point Running Average Filter • The LTI Running average FIR system is defined as : • Thus, the frequency response can be written as : • We can derive the magnitude and phase of the system by making use of the series expansion formula : … (5.3)

10. By letting  = e-jw, we can expand the frequency response, such that : • Now, • Where DL(w) is a well known function known as the ‘Dirichlet function’, where (L) is the order of the L-point running average filter. ( ) ( ) ( ) ( ) … (6.4)

11. A Closer Look at the Dirichlet Function • Consider what the frequency response would be for an 11-point running averager. • Thus, H(w) is a product of the real amplitude function D11(w) and a complex exponential function e-j5w. • ( Remember, e-j5w has magnitude = 1 and phase = -5w ) • ‘Amplitude’ rather than ‘Magnitude’ is used to describe D11(w) because D11(w) can be –ve. • We obtain a plot of the magnitude |H(w)| by taking the absolute value of D11(w). • We shall consider the amplitude representation first because it is simpler to examine the properties of the amplitude. ( )

12. The amplitude plot of the 11-point running averager is shown below : • Important Features to note : • D11(w) is periodic with period 2. • D11(w) has a maximum value = 1, at w = 0. • D11(w) decays as (w) increases, with smallest nonzero amplitude at • w =   • D11(w) has zeros at nonzero multiples of 2/11 • ( In General, DL(w) has zeros at nonzero multiples of 2/L)

13. For completeness, we know the phase of the 11-point running averager is linear with gradient of –5w.

14. The Magnitude response • for the 11-point running averager is the absolute value of D11(w) : • |H(w)| = |D11(w)| • D11(w) has zeros at nonzero multiples of 2/11. • And null frequencies at these points • The phase response is : • More complicated than the linear function we saw before. • As we must include the algebraic sign in the phase function that the magnitude |H(w)| = |D11(w)| discards.

15. A closeup of one period shows, the phase has a discontinuity at every nulled frequency and is linear inbetween each discontinuity.

16. Phase jump of - at each sign change for –ve w Gradient = (L-1)/2 Phase jump of + at each sign change for +ve w • Moreover, in the amplitude we see that the discontinuities in the phase occur where the sign of the Dirichlet function changes. • At each sign change, where (w) is +ve we have a + phase jump. • At each sign change, where (w) is -ve we have a - phase jump. • Thus, we can construct the phase from gradient & phase jump knowledge.