# Discussion on Formula Sheet

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2 of 15
Question 1 (4 marks) Convert the complex-valued expression π§ = (1 β πβ3)6 into a polar form of π§ = ππππ where π > 0 and β π < π β€ π. Remember to show all step-by-step working.
Solution π = _____________________________ , π = _____________________________
3 of 15
Question 2 (4 marks) Suppose an analog signal is defined as π₯(π‘) = 4 cos (20ππ‘ + π 2 ) + 3 cos(20ππ‘) . By using phasor addition, simplify the above expression of π₯(π‘) into the standard form of π₯(π‘) = π΄ cos(ππ‘ + π) where π΄ > 0, π > 0 and β π < π β€ π.
Solution π΄ = __________________ , π = _______________________ , π = ____________________
4 of 15
Question 3 (5 marks) A signal π₯(π‘) has the two-sided spectrum representation as shown below. (a) Write an equation π₯(π‘) in sinusoidal form. (b) Is π₯(π‘) a periodic signal? If so, calculate its period?
Solution
5 of 15
Question 4 (5 marks) Suppose tx )( is a chirp signal defined as π₯(π‘) = ππ{πππ(π‘)} = cos(2π[πΌπ‘2 + π½π‘ + π]) . Calculate the values of ο‘ , ο’ and ο¦ so that the instantaneous frequency of tx )( will start at 3800 Hz and end at 800 Hz over the time interval 0 β€ π‘ β€ 3 seconds.
Solution πΌ = _________________ , π½ = _____________________ , π = _______________________
6 of 15
Question 5 (5 marks) Suppose that a discrete-time signal x[n] is defined as π₯[π] = 10 cos (0.2ππ β π 8) , and that it was obtained by sampling a continuous-time signal at a sampling rate of 800 samples per second. (a) Determine two different continuous-time signals x1(t) and x2(t) whose samples are equal to x[n]. Both of these signals should have a frequency within 0-800 Hz. Give a formula for each signal. (b) If x[n] is given by the equation above, determine the signal π¦(π‘) that will be reconstructed by an ideal D-to-C converter operating at a sampling rate of 1600 samples per second, as shown below.
Solution π₯
1(π‘) = ____________________________________________ π₯
2(π‘) = ____________________________________________ π¦(π‘) = _____________________________________________
D-to-C
π₯[π] π¦(π‘)
ππ
= 1/π π
7 of 15
Question 6 (3 marks) Suppose that the following three sub-systems are connected in cascade (i.e. in series to each other) to form an overall system π
1: π¦1[π] = π₯1[π] β π₯1 [π β 3] π
2: π¦2[π] = π₯2[π] + π₯2 [π β 2] π
3: π¦3[π] = π₯3[π β 1] + π₯3 [π β 2] Determine the impulse response β[π] of the overall system by using the polynomial multiplication technique.
Solution β[π] = ____________________________________________
8 of 15
Question 7 (4 marks) Suppose a discrete-time LTI system is described as π¦[π] = 3π₯[π β 1] β 2π₯[π β 2] + 4π₯[π β 3] . Draw the implementation of this system as a block diagram in the direct form as well as the
transposed form. Ensure that all necessary labels are provided.
Solution
Direct Form
Transposed Form
9 of 15
Question 8 (4 marks) For a particular LTI system, when the input is 1 ο½ nunx ][][ , the corresponding output is
1 ο€ο€ο€ nnnny ο­ο­ο­ο«ο½ ]2[3]1[2][][ . Determine the output 2 ny ][ when the input to the LTI system is 2 nununx ο­ο­ο½ ]4[2][3][ . Give your answer as a formula expressing 2 ny ][ in terms of known sequences, or give a list of values for n ο₯οΌοΌο₯ο­ .
Solution π¦2[π] = ____________________________________________
10 of 15
Question 9 (4 marks) The frequency response of a linear time-invariant filter is defined as
Hο¨ο· Λο© ο½ ο¨1ο« eο­ jο· Λ ο©ο¨1ο­ e j2ο° / 3eο­ jο· Λ ο©ο¨1ο­ eο­ j2ο° / 3eο­ jο· Λ ο©. Find the output signal π¦[π] when the input signal π₯[π] is a unit impulse.
Solution π¦[π] = _____________________________________________________________________
11 of 15
Question 10 (4 marks) A digital LTI system is described as π¦[π] = 2π₯[π] β 3π₯[π β 1] + 2π₯[π β 2]. Determine the frequency response π»(π Μ) of the above system. Express your answer in a polar form (magnitude and phase), i.e. π»(π Μ) = |π»(π Μ)|ππβ π»(π Μ ).
Solution |π»(π Μ)| = ____________________________________________ β π»(π Μ) = ____________________________________________
12 of 15
Question 11 (4 marks) A LTI system has system function π»(π§) = (1 + π§β2)(4 β π§β2). The input to this system is: π₯[π] = 10 + 15 cos (π 4 π β π 6) for β β < π < β. Determine the output of the system y[n] corresponding to the above input x[n]. Give an equation for y[n] that is valid for all n.
Solution π¦[π] = ____________________________________________
13 of 15
Question 12 (4 marks) Suppose an IIR filter is defined as π¦[π] = β0.5π¦[π β 1] + 2π₯[π] . Plot the pole-zero diagram for this filter in the z-plane. Ensure that all necessary labels are provided.
Solution
END OF EXAMINATION PAPER
2 of 15
Question 1 (5 marks) From the plot of the sinusoid π₯(π‘) versus π‘ below, determine with high accuracy the numerical values for the amplitude π΄, frequency π and phase π needed in the representation: π₯(π‘) = π΄ cos(ππ‘ + π), where π΄ > 0, π > 0, and βπ < π β€ π .
Solution π΄ = __________________ , π = _______________________ , π = ____________________
3 of 15
Question 2 (5 marks) Suppose an analog signal π₯(π‘) is defined as π₯(π‘) = 3 cos (20ππ‘ + π 4) + 6 sin(20ππ‘). By using phasor addition, simplify the above expression of π₯(π‘) into the standard form of π₯(π‘) = π΄ cos(ππ‘ + π), where π΄ > 0, π > 0, and βπ < π β€ π .
Solution π΄ = _____________ , π = ________________________ , π = _____________________
4 of 15
Question 3 (4 marks) Sketch the spectrum (magnitude and phase separately) of π₯(π‘) = cos2 (200π π‘ β π 6). Ensure proper labels are in place and the magnitude spectrum is positive-valued.
Solution
5 of 15
Question 4 (4 marks) An amplitude modulated (AM) cosine signal is defined as
οΆ οΈ
ο¦ο§ο¨
ο½ ο« ο«
2
x(t) 2 sin ο° t cos 13ο° t ο° . Determine the minimum sampling rate that can be used to sample x(t) without any aliasing.
Solution
6 of 15
Question 5 (5 marks) Suppose that a discrete-time signal x[n] is defined as π₯[π] = 10 cos (0.2ππ β π 7) , and that it was obtained by sampling a continuous-time signal at a sampling rate of 800 samples per second. (a) Determine two different continuous-time signals x1(t) and x2(t) whose samples are equal to x[n]. Both of these signals should have a frequency within 0-800 Hz. Give a formula for each signal. (b) If x[n] is given by the equation above, determine the signal π¦(π‘) that will be reconstructed by an ideal D-to-C converter operating at a sampling rate of 1600 samples per second, as shown below.
Solution π₯
1(π‘) = ____________________________________________ π₯
2(π‘) = ____________________________________________ π¦(π‘) = _____________________________________________
D-to-C
π₯[π] π¦(π‘)
ππ
= 1/π π
7 of 15
Question 6 (3 marks) Suppose that the following three sub-systems are connected in cascade (i.e. in series to each other) to form an overall system π
1: π¦1[π] = π₯1[π] β π₯1 [π β 1] π
2: π¦2[π] = π₯2[π] + π₯2 [π β 2] π
3: π¦3[π] = π₯3[π β 1] + π₯3 [π β 2] Determine the impulse response β[π] of the overall system by using the polynomial multiplication technique.
Solution β[π] = ____________________________________________
8 of 15
Question 7 (4 marks) Suppose a discrete-time LTI system is described as π¦[π] = 10π₯[π] β 2π₯[π β 1] + 4π₯[π β 3] . Draw the implementation of this system as a block diagram in the direct form as well as the
transposed form. Ensure that all necessary labels are provided.
Solution
Direct Form
Transposed Form
9 of 15
Question 8 (4 marks) For a particular LTI system, when the input is 1 ο½ nunx ][][ , the corresponding output is
1 ο€ο€ο€ nnnny ο­ο­ο­ο«ο½ ]3[5]1[2][][ . Determine the output 2 ny ][ when the input to the LTI system is 2 nununx ο­ο­ο½ ]2[6][3][ . Give your answer as a formula expressing 2 ny ][ in terms of known sequences, or give a list of values for n ο₯οΌοΌο₯ο­ .
Solution π¦2[π] = ____________________________________________
10 of 15
Question 9 (4 marks) The frequency response of a linear time-invariant filter is defined as π»(π Μ) = (1 + πβππ Μ ) (1 β πππ 3 πβππ Μ ) (1 β πβππ 3 πβππ Μ ) Find the output signal π¦[π] when the input signal π₯[π] is a unit impulse.
Solution π¦[π] = __________________________________________________________________
11 of 15
Question 10 (4 marks) A digital LTI system is described as π¦[π] = 5π₯[π] + 3π₯[π β 1] + 10π₯[π β 4]. Determine the frequency response π»(π Μ) of the above system. Express your answer in a polar form (magnitude and phase), i.e. π»(π Μ) = |π»(π Μ)|ππβ π»(π Μ ).
Solution |π»(π Μ)| = ____________________________________________ β π»(π Μ) = ____________________________________________
12 of 15
Question 11 (4 marks) Suppose that a digital system is defined by its system function as π»(π§) = (1 β π§β1)(1 + π§β2)(1 + π§β1) . Write the time-domain description of this system in the form of a difference equation.
Solution π¦[π] = ____________________________________________
13 of 15
Question 12 (4 marks) Suppose an IIR filter is defined as π¦[π] = π¦[π β 1] β π¦[π β 3] + π₯[π] . Plot the pole-zero diagram for this filter in the z-plane. Ensure that all necessary labels are provided.
Solution
END OF EXAMINATION PAPER
15 of 15
FORMULA SHEET
If ππ₯2 + ππ₯ + π = 0 then the root π₯1,2 = βπ Β± βπ2β4ππ
2π
Sinusoidal Signals:
sin(π) = cos(π β π/2) sin(βπ) = β sin(π)
cos(π) = sin(π + π/2) cos(βπ) = cos(π)
Sinc Function:
sinc (π₯) =
sin(ππ₯)
ππ₯
Eulerβs Formula:
πππ = cos π + π sin π
πβππ = cos π β π sin π
cos π =
πππ + πβππ
2
sin π =
πππ β πβππ
2π
Sum of Geometric Series:
β πΌπ
πΏβ1
π=0
=
1 β πΌπΏ
1 β πΌ
Conversion from Analog to Digital Signal:
π₯(π‘) β π₯[π] = π₯(πππ )
For FIR Filter with Coefficients {ππ}:
ο· Difference Equation:
π¦[π] = β β[π]π₯[π β π]
π
π=0
= β πππ₯[π β π]
π
π=0
ο· Impulse Response:
β[π] = β β[π]πΏ[π β π]
π
π=0
= β πππΏ[π β π]
π
π=0
ο· Frequency Response:
π»(π Μ) = β β[π]πβππ Μ π
π
π=0
= β πππβππ Μ π
π
π=0
ο· System Function:
π»(π§) = β β[π]π§βπ
π
π=0
= β πππ§βπ
π
π=0
Z-transform:
π(π§) = β π₯[π]π§βπ
π
π=0
π₯[π β π] β π§βππ(π§)
– End of Formula Sheet –

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