Assignment on mathematics

please have strong pain point for each body paragraph and 5-6 sentences in each paragraph.

2 of 15
Question 1 (4 marks) Convert the complex-valued expression 𝑧 = (1 βˆ’ π‘—βˆš3)6 into a polar form of 𝑧 = π‘Ÿπ‘’π‘—πœƒ where π‘Ÿ > 0 and – πœ‹ < πœƒ ≀ πœ‹. Remember to show all step-by-step working.
Solution π‘Ÿ = _____________________________ , πœƒ = _____________________________
3 of 15
Question 2 (4 marks) Suppose an analog signal is defined as π‘₯(𝑑) = 4 cos (20πœ‹π‘‘ + πœ‹ 2 ) + 3 cos(20πœ‹π‘‘) . By using phasor addition, simplify the above expression of π‘₯(𝑑) into the standard form of π‘₯(𝑑) = 𝐴 cos(πœ”π‘‘ + πœ™) where 𝐴 > 0, πœ” > 0 and – πœ‹ < πœ™ ≀ πœ‹.
Solution 𝐴 = __________________ , πœ” = _______________________ , πœ™ = ____________________
4 of 15
Question 3 (5 marks) A signal π‘₯(𝑑) has the two-sided spectrum representation as shown below. (a) Write an equation π‘₯(𝑑) in sinusoidal form. (b) Is π‘₯(𝑑) 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‘₯(𝑑) = 𝑅𝑒{π‘’π‘—πœƒ(𝑑)} = cos(2πœ‹[𝛼𝑑2 + 𝛽𝑑 + πœ™]) . Calculate the values of  ,  and  so that the instantaneous frequency of tx )( will start at 3800 Hz and end at 800 Hz over the time interval 0 ≀ 𝑑 ≀ 3 seconds.
Solution 𝛼 = _________________ , 𝛽 = _____________________ , πœ™ = _______________________
6 of 15
Question 5 (5 marks) Suppose that a discrete-time signal x[n] is defined as π‘₯[𝑛] = 10 cos (0.2πœ‹π‘› βˆ’ πœ‹ 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 𝑦(𝑑) 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(𝑑) = ____________________________________________ π‘₯
2(𝑑) = ____________________________________________ 𝑦(𝑑) = _____________________________________________
D-to-C
π‘₯[𝑛] 𝑦(𝑑)
𝑇𝑠
= 1/𝑓 𝑠
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 𝑆
1: 𝑦1[𝑛] = π‘₯1[𝑛] βˆ’ π‘₯1 [𝑛 βˆ’ 3] 𝑆
2: 𝑦2[𝑛] = π‘₯2[𝑛] + π‘₯2 [𝑛 βˆ’ 2] 𝑆
3: 𝑦3[𝑛] = π‘₯3[𝑛 βˆ’ 1] + π‘₯3 [𝑛 βˆ’ 2] Determine the impulse response β„Ž[𝑛] of the overall system by using the polynomial multiplication technique.
Solution β„Ž[𝑛] = ____________________________________________
8 of 15
Question 7 (4 marks) Suppose a discrete-time LTI system is described as 𝑦[𝑛] = 3π‘₯[𝑛 βˆ’ 1] βˆ’ 2π‘₯[𝑛 βˆ’ 2] + 4π‘₯[𝑛 βˆ’ 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  nnnny  ]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 𝑦2[𝑛] = ____________________________________________
10 of 15
Question 9 (4 marks) The frequency response of a linear time-invariant filter is defined as
H ˆ ο€½ 1 eο€­ j Λ† 1ο€­ e j2 / 3eο€­ j Λ† 1ο€­ eο€­ j2 / 3eο€­ j Λ† . Find the output signal 𝑦[𝑛] when the input signal π‘₯[𝑛] is a unit impulse.
Solution 𝑦[𝑛] = _____________________________________________________________________
11 of 15
Question 10 (4 marks) A digital LTI system is described as 𝑦[𝑛] = 2π‘₯[𝑛] βˆ’ 3π‘₯[𝑛 βˆ’ 1] + 2π‘₯[𝑛 βˆ’ 2]. Determine the frequency response 𝐻(πœ” Μ‚) of the above system. Express your answer in a polar form (magnitude and phase), i.e. 𝐻(πœ” Μ‚) = |𝐻(πœ” Μ‚)|π‘’π‘—βˆ π»(πœ” Μ‚ ).
Solution |𝐻(πœ” Μ‚)| = ____________________________________________ ∠𝐻(πœ” Μ‚) = ____________________________________________
12 of 15
Question 11 (4 marks) A LTI system has system function 𝐻(𝑧) = (1 + π‘§βˆ’2)(4 βˆ’ π‘§βˆ’2). The input to this system is: π‘₯[𝑛] = 10 + 15 cos (πœ‹ 4 𝑛 βˆ’ πœ‹ 6) for βˆ’ ∞ < 𝑛 < ∞. 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 𝑦[𝑛] = ____________________________________________
13 of 15
Question 12 (4 marks) Suppose an IIR filter is defined as 𝑦[𝑛] = βˆ’0.5𝑦[𝑛 βˆ’ 1] + 2π‘₯[𝑛] . 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‘₯(𝑑) versus 𝑑 below, determine with high accuracy the numerical values for the amplitude 𝐴, frequency πœ” and phase πœ™ needed in the representation: π‘₯(𝑑) = 𝐴 cos(πœ”π‘‘ + πœ™), where 𝐴 > 0, πœ” > 0, and βˆ’πœ‹ < πœ™ ≀ πœ‹ .
Solution 𝐴 = __________________ , πœ” = _______________________ , πœ™ = ____________________
3 of 15
Question 2 (5 marks) Suppose an analog signal π‘₯(𝑑) is defined as π‘₯(𝑑) = 3 cos (20πœ‹π‘‘ + πœ‹ 4) + 6 sin(20πœ‹π‘‘). By using phasor addition, simplify the above expression of π‘₯(𝑑) into the standard form of π‘₯(𝑑) = 𝐴 cos(πœ”π‘‘ + πœ™), where 𝐴 > 0, πœ” > 0, and βˆ’πœ‹ < πœ™ ≀ πœ‹ .
Solution 𝐴 = _____________ , πœ” = ________________________ , πœ™ = _____________________
4 of 15
Question 3 (4 marks) Sketch the spectrum (magnitude and phase separately) of π‘₯(𝑑) = cos2 (200πœ‹ 𝑑 βˆ’ πœ‹ 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
   οƒ·
οƒΆ οƒΈ

ο€½  
2
x(t) 2 sin  t cos 13 t  . 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‘₯[𝑛] = 10 cos (0.2πœ‹π‘› βˆ’ πœ‹ 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 𝑦(𝑑) 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(𝑑) = ____________________________________________ π‘₯
2(𝑑) = ____________________________________________ 𝑦(𝑑) = _____________________________________________
D-to-C
π‘₯[𝑛] 𝑦(𝑑)
𝑇𝑠
= 1/𝑓 𝑠
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 𝑆
1: 𝑦1[𝑛] = π‘₯1[𝑛] βˆ’ π‘₯1 [𝑛 βˆ’ 1] 𝑆
2: 𝑦2[𝑛] = π‘₯2[𝑛] + π‘₯2 [𝑛 βˆ’ 2] 𝑆
3: 𝑦3[𝑛] = π‘₯3[𝑛 βˆ’ 1] + π‘₯3 [𝑛 βˆ’ 2] Determine the impulse response β„Ž[𝑛] of the overall system by using the polynomial multiplication technique.
Solution β„Ž[𝑛] = ____________________________________________
8 of 15
Question 7 (4 marks) Suppose a discrete-time LTI system is described as 𝑦[𝑛] = 10π‘₯[𝑛] βˆ’ 2π‘₯[𝑛 βˆ’ 1] + 4π‘₯[𝑛 βˆ’ 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  nnnny  ]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 𝑦2[𝑛] = ____________________________________________
10 of 15
Question 9 (4 marks) The frequency response of a linear time-invariant filter is defined as 𝐻(πœ” Μ‚) = (1 + π‘’βˆ’π‘—πœ” Μ‚ ) (1 βˆ’ π‘’π‘—πœ‹ 3 π‘’βˆ’π‘—πœ” Μ‚ ) (1 βˆ’ π‘’βˆ’π‘—πœ‹ 3 π‘’βˆ’π‘—πœ” Μ‚ ) Find the output signal 𝑦[𝑛] when the input signal π‘₯[𝑛] is a unit impulse.
Solution 𝑦[𝑛] = __________________________________________________________________
11 of 15
Question 10 (4 marks) A digital LTI system is described as 𝑦[𝑛] = 5π‘₯[𝑛] + 3π‘₯[𝑛 βˆ’ 1] + 10π‘₯[𝑛 βˆ’ 4]. Determine the frequency response 𝐻(πœ” Μ‚) of the above system. Express your answer in a polar form (magnitude and phase), i.e. 𝐻(πœ” Μ‚) = |𝐻(πœ” Μ‚)|π‘’π‘—βˆ π»(πœ” Μ‚ ).
Solution |𝐻(πœ” Μ‚)| = ____________________________________________ ∠𝐻(πœ” Μ‚) = ____________________________________________
12 of 15
Question 11 (4 marks) Suppose that a digital system is defined by its system function as 𝐻(𝑧) = (1 βˆ’ π‘§βˆ’1)(1 + π‘§βˆ’2)(1 + π‘§βˆ’1) . Write the time-domain description of this system in the form of a difference equation.
Solution 𝑦[𝑛] = ____________________________________________
13 of 15
Question 12 (4 marks) Suppose an IIR filter is defined as 𝑦[𝑛] = 𝑦[𝑛 βˆ’ 1] βˆ’ 𝑦[𝑛 βˆ’ 3] + π‘₯[𝑛] . 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
Quadratic Roots:
If π‘Žπ‘₯2 + 𝑏π‘₯ + 𝑐 = 0 then the root π‘₯1,2 = βˆ’π‘ Β± βˆšπ‘2βˆ’4π‘Žπ‘
2π‘Ž
Sinusoidal Signals:
sin(πœƒ) = cos(πœƒ βˆ’ πœ‹/2) sin(βˆ’πœƒ) = βˆ’ sin(πœƒ)
cos(πœƒ) = sin(πœƒ + πœ‹/2) cos(βˆ’πœƒ) = cos(πœƒ)
Sinc Function:
sinc (π‘₯) =
sin(πœ‹π‘₯)
πœ‹π‘₯
Euler’s Formula:
π‘’π‘—πœƒ = cos πœƒ + 𝑗 sin πœƒ
π‘’βˆ’π‘—πœƒ = cos πœƒ βˆ’ 𝑗 sin πœƒ
cos πœƒ =
π‘’π‘—πœƒ + π‘’βˆ’π‘—πœƒ
2
sin πœƒ =
π‘’π‘—πœƒ βˆ’ π‘’βˆ’π‘—πœƒ
2𝑗
Sum of Geometric Series:
βˆ‘ π›Όπ‘˜
πΏβˆ’1
π‘˜=0
=
1 βˆ’ 𝛼𝐿
1 βˆ’ 𝛼
Conversion from Analog to Digital Signal:
π‘₯(𝑑) β†’ π‘₯[𝑛] = π‘₯(𝑛𝑇𝑠)
For FIR Filter with Coefficients {π’ƒπ’Œ}:
ο‚· Difference Equation:
𝑦[𝑛] = βˆ‘ β„Ž[π‘˜]π‘₯[𝑛 βˆ’ π‘˜]
𝑀
π‘˜=0
= βˆ‘ π‘π‘˜π‘₯[𝑛 βˆ’ π‘˜]
𝑀
π‘˜=0
ο‚· Impulse Response:
β„Ž[𝑛] = βˆ‘ β„Ž[π‘˜]𝛿[𝑛 βˆ’ π‘˜]
𝑀
π‘˜=0
= βˆ‘ π‘π‘˜π›Ώ[𝑛 βˆ’ π‘˜]
𝑀
π‘˜=0
ο‚· Frequency Response:
𝐻(πœ” Μ‚) = βˆ‘ β„Ž[π‘˜]π‘’βˆ’π‘—πœ” Μ‚ π‘˜
𝑀
π‘˜=0
= βˆ‘ π‘π‘˜π‘’βˆ’π‘—πœ” Μ‚ π‘˜
𝑀
π‘˜=0
ο‚· System Function:
𝐻(𝑧) = βˆ‘ β„Ž[π‘˜]π‘§βˆ’π‘˜
𝑀
π‘˜=0
= βˆ‘ π‘π‘˜π‘§βˆ’π‘˜
𝑀
π‘˜=0
Z-transform:
𝑋(𝑧) = βˆ‘ π‘₯[π‘˜]π‘§βˆ’π‘˜
𝑀
π‘˜=0
π‘₯[𝑛 βˆ’ π‘˜] ↔ π‘§βˆ’π‘˜π‘‹(𝑧)
– End of Formula Sheet –

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