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Question 1 (4 marks) Convert the complex-valued expression π§ = (1 β πβ3)6 into a polar form of π§ = ππππ where π > 0 and β π < π β€ π. Remember to show all step-by-step working.

Solution π = _____________________________ , π = _____________________________

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Question 2 (4 marks) Suppose an analog signal is defined as π₯(π‘) = 4 cos (20ππ‘ + π 2 ) + 3 cos(20ππ‘) . By using phasor addition, simplify the above expression of π₯(π‘) into the standard form of π₯(π‘) = π΄ cos(ππ‘ + π) where π΄ > 0, π > 0 and β π < π β€ π.

Solution π΄ = __________________ , π = _______________________ , π = ____________________

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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

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Question 4 (5 marks) Suppose *tx *)( is a chirp signal defined as π₯(π‘) = π
π{πππ(π‘)} = cos(2π[πΌπ‘2 + π½π‘ + 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 πΌ = _________________ , π½ = _____________________ , π = _______________________

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Question 5 (5 marks) Suppose that a discrete-time signal *x*[*n*] is defined as π₯[π] = 10 cos (0.2ππ 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 *x*1(*t*) and *x*2(*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‘)

ππ

= 1/π π

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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] = ____________________________________________

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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

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Question 8 (4 marks) For a particular LTI system, when the input is 1 ο½ *nunx *][][ , the corresponding output is

1 ο€ο€ο€ *nnnny *οοοο«ο½ ]2[3]1[2][][ . Determine the output 2 *ny *][ when the input to the LTI system is 2 *nununx *οοο½ ]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₯οΌοΌο₯ο .

Solution π¦2[π] = ____________________________________________

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Question 9 (4 marks) The frequency response of a linear time-invariant filter is defined as

*H*ο¨ο· Λο© ο½ ο¨1ο« *e*ο *j*ο· Λ ο©ο¨1ο *e **j*2ο° / 3*e*ο *j*ο· Λ ο©ο¨1ο *e*ο *j*2ο° / 3*e*ο *j*ο· Λ ο©. Find the output signal π¦[π] when the input signal π₯[π] is a unit impulse.

Solution π¦[π] = _____________________________________________________________________

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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)|ππβ π»(π Μ ).

Solution |π»(π Μ)| = ____________________________________________ β π»(π Μ) = ____________________________________________

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Question 11 (4 marks) A LTI system has system function π»(π§) = (1 + π§β2)(4 β π§β2). 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] = ____________________________________________

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Question 12 (4 marks) Suppose an IIR filter is defined as π¦[π] = β0.5π¦[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(ππ‘ + π), where π΄ > 0, π > 0, and βπ < π β€ π .

Solution π΄ = __________________ , π = _______________________ , π = ____________________

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Question 2 (5 marks) Suppose an analog signal π₯(π‘) is defined as π₯(π‘) = 3 cos (20ππ‘ + π 4) + 6 sin(20ππ‘). By using phasor addition, simplify the above expression of π₯(π‘) into the standard form of π₯(π‘) = π΄ cos(ππ‘ + π), where π΄ > 0, π > 0, and βπ < π β€ π .

Solution π΄ = _____________ , π = ________________________ , π = _____________________

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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 ο¨ ο©ο ο·

οΆ οΈ

ο¦ο§ο¨

ο½ 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 (0.2ππ 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 *x*1(*t*) and *x*2(*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‘)

ππ

= 1/π π

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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 ο½ *nunx *][][ , the corresponding output is

1 ο€ο€ο€ *nnnny *οοοο«ο½ ]3[5]1[2][][ . Determine the output 2 *ny *][ when the input to the LTI system is 2 *nununx *οοο½ ]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₯οΌοΌο₯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 + πβππ Μ ) (1 β πππ 3 πβππ Μ ) (1 β πβππ 3 πβππ Μ ) Find the output signal π¦[π] when the input signal π₯[π] is a unit impulse.

Solution π¦[π] = __________________________________________________________________

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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)|ππβ π»(π Μ ).

Solution |π»(π Μ)| = ____________________________________________ β π»(π Μ) = ____________________________________________

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Question 11 (4 marks) Suppose that a digital system is defined by its system function as π»(π§) = (1 β π§β1)(1 + π§β2)(1 + π§β1) . Write the time-domain description of this system in the form of a difference equation.

Solution π¦[π] = ____________________________________________

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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

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FORMULA SHEET

Quadratic Roots:

If ππ₯2 + ππ₯ + π = 0 then the root π₯1,2 = βπ Β± βπ2β4ππ

2π

Sinusoidal Signals:

sin(π) = cos(π β π/2) sin(βπ) = β sin(π)

cos(π) = sin(π + π/2) cos(βπ) = cos(π)

Sinc Function:

sinc (π₯) =

sin(ππ₯)

ππ₯

Eulerβs Formula:

πππ = cos π + π sin π

πβππ = cos π β π sin π

cos π =

πππ + πβππ

2

sin π =

πππ β πβππ

2π

Sum of Geometric Series:

β πΌπ

πΏβ1

π=0

=

1 β πΌπΏ

1 β πΌ

Conversion from Analog to Digital Signal:

π₯(π‘) β π₯[π] = π₯(πππ )

For FIR Filter with Coefficients {ππ}:

ο· Difference Equation:

π¦[π] = β β[π]π₯[π β π]

π

π=0

= β πππ₯[π β π]

π

π=0

ο· Impulse Response:

β[π] = β β[π]πΏ[π β π]

π

π=0

= β πππΏ[π β π]

π

π=0

ο· Frequency Response:

π»(π Μ) = β β[π]πβππ Μ π

π

π=0

= β πππβππ Μ π

π

π=0

ο· System Function:

π»(π§) = β β[π]π§βπ

π

π=0

= β πππ§βπ

π

π=0

Z-transform:

π(π§) = β π₯[π]π§βπ

π

π=0

π₯[π β π] β π§βππ(π§)

– End of Formula Sheet –

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