1.Making cocktails with the Z-transform:The very famous Fibonacci sequence goes like this:0,1,1,2,3,5,8,13,21,34,55,89The magical and mystical ‘golden ratio’φ= 1.618033988749895 is the limiting value of the ratiobetween two terms in the sequence eg.,5534= 1.61765,8955= 1.6181818The Fibonacci sequence is generated by the following difference-equation:y(k+ 2) =y(k+ 1) +y(k)k= 0,1,2,…(1)withy(0) = 0,y(1) = 1 as initial conditions. Here each term in the sequence is the sum of the twoprevious. Thusy(2) =y(1) +y(0), y(3) =y(2) +y(1), y(4) =y(3) +y(2)etc.Given that you know the difference equation that generates the Fibonacci sequence, use Z-transformsto calculatey(k) andφ, as set out in the following steps.(a) Using the difference-equation above, show that the Z-transformY(z) of the Fibonacci sequencey(k) is:Y(z) =zz2−z−1(b) Show thatY(z) can be equivalently written as:Y(z) =z(z−α)(z−β)whereα=φ=1+√52= 1.618033988749895,known as the golden ratio,andβ= 1−φ.[Hint: Factorizez2−z−1 by hypothetically lettingz2−z−1 = 0 and then finding theαandβyou need so thatz2−z−1 = (z−α)(z−β) = 0. That is you use the ’formula’ to find rootsofz2−z−1 = 0.](c) Find the partial fraction decomposition for1zY(z) =1(z−α)(z−β)=Az−α+Bz−βThat is, show thatA=φ/√5, and find B.(d) Use the inverse Z-transform to find an exact expression fory(k), namelyy(k) =φk√5−(1−φ)k√5.(e) Find the first five terms in the sequence from the solution you found above fory(k) in (d) andshow that it conforms to the solution expected of the difference equation Eqn.1.
(f) Showy(k)≈φk√5for largek.(g) Hence showy(k+ 1)y(k)≈φ(h) Why is the golden ratio considered magical or mystical? Relate this to physical quantities suchas the size of the pyramids, the Mona Lisa, pine cones, door frames, cocktails (see attached),and other pleasing structural ratios etc.(i) Why might it be incorrect to refer to the numbers 0,1,1,2,3,5,8.13… as the Fibonacci series?(a=3, b=2, c=3, d=3, e=2, f=2, g=2, h=2, i=1. Total =20 marks)2.Fourier TransformA beaker of water of temperaturey(t) at timetis cooling to room temperature. To keep thingssimple and without loss of generality, we will suppose room temperatute is 0 degrees centigrade.The temperature of the watery(t) satisfies Newton’s Law of cooling which states:dy(t)dt=−ky(t) +x(t)Herex(t) are external temperature fluctuations that impact the cooling process of the beaker. Onewonders how these fluctuations influence the cooling process. We want to study the frequencyresponse of this system to the forcing fluctuationsx(t).
(a) DenoteY(ω) andX(ω) as the Fourier transforms ofy(t) andx(t) respectively.By taking Fourier transforms of the above equation, and giving working, show thatY(ω) is ofthe form:Y(ω) =ab+jcX(ω)Find a,b, and c.(b) For a general input, the frequency response of the system is given byH(ω) =Y(ω)X(ω). FindH(ω).(c) How does one obtainH(ω) in practice? [This is not an assessment question!] One has toexamine the system for all different frequency inputsω. It might be possible to test onefrequencyω0after another and thus sweepωto obtain a picture ofH(ω) for allω.A neater solution requires only pulsing the system with a large impulse that can be modelledby the delta functionx(t) =δ(t). This will give us the so called impulse responseH(ω) (orfrequency response) to a delta signal.Why do this? Recall that the Fourier Transform ofx(t) =δ(t) isF(δ(t)) =X(ω) = 1. Thismeans that the impulsex(t) =δ(t) has energy for any and all frequency componentsω. So asingle pulsex(t) =δ(t) will automatically yield the spectrumY(ω) =H(ω)X(ω) =H(ω) forallω.Now that you have calculatedH(ω) for the beaker system, separate it into its real and imag-inary components i.e., in the formH(ω) =a+jb.Find alegebraic expressions for themagnitude|H(ω)|and phaseφ(ω)for this system.(d) Sketch a graph of|H(ω)|as a function ofω.
(e) Provide a careful interpretation of this graph. What does it mean in terms of the inputfluctuationsx(t) (eg., anx(t) that is dominated by high frequencies as compared to anx(t)that is just a low frequency oscillation).
(f) Suppose the inputx(t) =e−tu(t). FindY(ω).(g) Take the inverse Fourier transform ofY(ω) and findy(t). Sketchy(t).
(h) What is the solution of the above differential equation for cooling whenx(t) = 0. Sketch this solution on the same axes as above and compare with your last plot. Explain what you observe.(a=2, b=1, c=4, d=1, e=2, f=4, g=4, h=2. Total =20 marks)
3. Without using tables (i.e., from first principles) find the Fourier transform of:y(t) =etu(t+ 1) +e3tu(t−3).(7 marks)
4. Find the inverse Fourier transform ofY(ω) =δ(ω−3). [Hint: Use a shift theorem]Can you give some intuitive interpret to your solutiony(k)?(7 marks)5. Find the inverse Z-transform ofY(z) =zz2−5z+ 6.(6 marks
