1. MM3FC Mathematical Modeling 3 LECTURE 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 Lecture What 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 b k ={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. 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. x[n] = e jwn H 1 [w]e jwn LTI 1 H 1 [w] LTI 2 H 2 [w] y 1 [n]= H 1 [w]H 2 [w]e jwn LTI 2 H 2 [w] LTI 1 H 1 [n] x[n] = e jwn H 2 [w]e jwn y 2 [n]= H 2 [w]H 1 [w]e jwn = H 1 [w]H 2 [w]e jwn LTI Equivalent H[w] x[n] = e jwn y[n] = H[w]e jwn

8. Thus, Example 1 : Two LTI systems have coefficients a k ={1,-2} and b k ={0,1,1}. Determine their cascaded frequency response, impulse response, difference equation and the co-efficients of an equivalent filter. H 1 (w) = 1 – 2e -jw and H 2 (w) = e -jw + e -2jw H(w) = H 1 (w)H 2 (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 : c k = {0,1,-1,-2} ( Quite handy if you have 3 or more cascaded systems) … (5.2)

9. 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 : Frequency Response of an L-point Running Average Filter … (5.3)

10. By letting  = e -jw, we can expand the frequency response, such that : Now, Where D L (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 D 11 (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 D 11 (w) because D 11 (w) can be –ve. We obtain a plot of the magnitude |H(w)| by taking the absolute value of D 11 (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 : 1)D 11 (w) is periodic with period 2 . 2)D 11 (w) has a maximum value = 1, at w = 0. 3)D 11 (w) decays as (w) increases, with smallest nonzero amplitude at w =   1)D 11 (w) has zeros at nonzero multiples of 2  /11 ( In General, D L (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 D 11 (w) : |H(w)| = |D 11 (w)| D 11 (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)| = |D 11 (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. 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. Phase jump of +  at each sign change for +ve w Phase jump of -  at each sign change for –ve w Gradient = (L-1)/2