DIT, Kevin St,
Signals and systems
S008_2
S008_2: Signals and Systems questions
Transfer functions
Q1
(a) Draw the circuit diagram and derive the transfer function, for each of the following first-order RC
filters:
(i) Low - pass
(ii) High - pass
Sketch the asymptotic response for each circuit. Discuss briefly the significance of each response.
Explain the term roll-off.
[9 marks]
(b) In (ii) above, determine the values of the resistors in the circuit if the two break frequencies are 10
kHz and 100 kHz and the capacitance used is 10 nF.
[11 marks]
Q2
(a)Derive an expression for the voltage transfer function for a modified low-pass CR filter.
[10marks]
(b)Hence plot the form of the asymptotic amplitude and phase response of the filter.
[10marks]
Q3
Derive the voltage transfer function H(s) and sketch the pole-zero plot for the network in figure 1.
[10 marks]
State the values of the two cut-off frequencies, hence sketch the asymptotic amplitude response.
[10 marks]
R1 = 85 kΩ, R2 = 15 kΩ, C = 10 nF.
Figure 1
Q4
Obtain the transfer function for the Wien network in figure 2. Hence sketch the asymptotic amplitude
response indicating clearly the frequency at which minimum attenuation occurs and the value of his
attenuation in dB.
1
Copyright Paul Tobin, School of Electronics and Communications Engineering
DIT, Kevin St,
Signals and systems
S008_2
Figure 2
Q5
Two low-pass CR networks are cascaded via a unity gain buffer amplifier. Obtain the transfer function
for this network and hence draw
a) The asymptotic amplitude response, and
(12 marks)
b) A polar plot (polar paper supplied).
(8 marks]
Q6
Show that the relationship between the input and output voltages is given as:
Vo
1 + jωτ
=
Vin 1 + jωτα
Where τ1 is CR2 and α is (R1 + R2)/R1
[10 marks]
Plot the asymptotic amplitude response for this transfer function. Calculate the phase shift through the
network at a frequency ω = 3000 r/s.
R1 = 90 kΩ, R2 = 10 kΩ, C = 0.01 uF.
[10 marks]
Q7
The output of a high-pass LR network is connected via an ideal unity gain buffer amplifier to an identical
high-pass LR network. Obtain the voltage transfer function for this system and hence sketch the
amplitude Bode response.
R = 1 kΩ, L = 100 mH.
[20marks]
Q8
Obtain the voltage transfer function for the network shown in figure 3.
[10 marks]
Hence by Bode analysis, plot the approximate frequency response showing the values of the break
frequencies.
R1 = 10 kΩ, R2 = 100 kΩ, C1 = C2 = 10 nF.
2
Copyright Paul Tobin, School of Electronics and Communications Engineering
DIT, Kevin St,
Signals and systems
S008_2
Figure 3
Q9
The magnitude of the voltage gain(The transfer function) of the circuit in figure 4 is:
R
Av =
Z
Where Z is a parallel connection of a resistance R and a capacitance C. You may assume the operational
amplifier is ideal. Determine the voltage transfer function. Sketch the asymptotic amplitude and phase
response and state the break frequency value.
Figure 4
Q10
Obtain the transfer function for the network in figure 5. Sketch the asymptotic amplitude response
indicating clearly the break frequency values and the attenuation in dB at DC. Give an expression for the
phase between the input and output voltages at a frequency f.
R1 = 20 kΩ, R2 =1 kΩ, L = 10 mH.
[20 marks]
Figure 5
Q11
A potential divider formed by two resistors R1 and R2 has an inductor connected across R2. Derive an
expression for the transfer function if the input voltage is applied to the divider network and the output
voltage developed across R2. Using this transfer function, sketch the form of the asymptotic response and
calculate the break frequency fl and a value for the attenuation a decade above fl given that the component
values are:
R1 = 20 kΩ, R2 = 1 kΩ, L = 1 mH.
[20 marks]
3
Copyright Paul Tobin, School of Electronics and Communications Engineering
DIT, Kevin St,
Signals and systems
S008_2
Figure 6
Q12
Determine the voltage transfer function for the circuit in figure 7. Sketch the asymptotic amplitude
response and state the break frequency values.
R = 10 kΩ, C = 10 nF
Figure 7
Q13
(a) Draw the circuit diagram, and derive the transfer function, for a simple first
order low-pass RC
filter. Sketch the asymptotic amplitude response. Explain the terms: cut-off frequency and roll-off rate.
[12 marks]
(b) Show how this circuit may be modified to ensure a constant attenuation at high frequencies.
Determine the component values for this modified circuit if the two cut-off frequencies are 10 kHz and
100 kHz using a 10 nF capacitor.
[8 marks]
Q14
(a) Explain the terms cut-off frequency and roll-off rate. Illustrate your answer using a simple CR lowpass filter.
[6 marks]
(b) The transfer function for a two-port network is
⎛ 1 + j ω ω c1 ⎞
⎟⎟
TF = ⎜⎜
⎝ 1 + j ω ωc2 ⎠
Where ω C1 = 100.103 rs -1 and ω C 2 = = 9.103 rs -1.
For the values given, plot the asymptotic amplitude response.
[14 marks]
Q15
Obtain the voltage transfer function for the network shown in figure 8.
[ 10 marks]
4
Copyright Paul Tobin, School of Electronics and Communications Engineering
DIT, Kevin St,
Signals and systems
S008_2
State the values of the break frequencies and hence plot the asymptotic amplitude response.
R1 = 10 kΩ , R2 = 1k Ω , C= 10nF.
[10 marks]
Figure 8
Thévenin Circuits
Q15
Obtain the Thévenin and Norton equivalent circuits for the circuit in Figure 10. Hence determine the
current in an impedance Z or the condition of maximum power transfer, if the load is formed by R3 and
L.
[12 marks]
Calculate a value for the current in R3 if the value for L is chosen as the complex conjugate of the
Thévenin impedance. You may assume the voltage source is ideal. E= 50 ∠50, R1 = R2 = R3 = 5 Ω, Xc =
2 Ω,
[8 marks]
Q16
For the circuit shown in figure 11, calculate, by developing an appropriate Thévenin equivalent circuit,
the value of R that will be required to achieve maximum power transfer into the resistor R. Obtain the
Norton equivalent circuit between the terminals A and B in figure 12. Hence calculate the current which
flows in an impedance Z = 10 +j0 Ω connected between A and B.
R1 = 30 Ω, R2 = 40 Ω, R3 = 10 Ω, R4 = 20 Ω, Xc1 = -j20 Ω, Xc2 = -j40 Ω, E = 10∠ 220.
[20 marks]
Q17
By developing an appropriate Thévenin equivalent circuit, calculate the values for R and X to achieve
maximum power transfer into the resistor R for the circuit shown in Figure 13. Using these values for R
and X and removing link CD, calculate the Z-parameters of the resulting 2-port network and hence draw
the Z-equivalent circuit.
R1 = R2 = 5 Ω; X1 = X2 = -j2 Ω
[20 marks]
Resonance
Q18
A tuned circuit consists of a 100 uH coil and a variable capacitance connected in parallel. The ac
resistance of the coil is 10 Ω. If the circuit is required to be resonant at 1 MHz. Calculate the dynamic
resistance, bandwidth and capacitance value. What is the value of the impedance and phase of the circuit
5
Copyright Paul Tobin, School of Electronics and Communications Engineering
DIT, Kevin St,
Signals and systems
S008_2
10 kHz off resonance? Calculate the loaded quality factor for the circuit in figure 1. The tuned circuit
components have the same values and the coefficient of coupling between the two section of the coil is
unity. Justify any approximations used in the solution. N1 = 10 T, N2 =10 T, rds = 10 kΩ.
Q19
Discuss the effects of source impedance on the selectivity of series and parallel tuned LCR circuit series
resistance of the inductor at the resistance R which when connected across will double the circuit
bandwidth, and the assumptions made in the above analysis
[14 marks]
Q20
Determine the resonant frequency for a parallel-tuned circuit. The inductance L has a value of 20 uH,
QUL = 25 and C = 10 nF. Calculate the value of R, which when placed across C, will produce a loaded Qfactor of 20. Hence determine the circuit bandwidth, and calculate the impedance at resonance.
Q21
Explain what is meant by the loaded Q-factor of a tuned circuit
[4marks]
A tuned RF amplifier is required to have a bandwidth of 100 kHz at a resonant frequency fo = 1 MHz.
The total resistive loading on the tuned circuit is 100 kΩ and the parasitic capacitance is 30 pF. Calculate
the resistance, at fo, of a 1 mH inductance used in this circuit and the capacitance required to achieve the
desired resonant frequency.
[16 marks]
Q22
Discuss the effects of source impedance on the selectivity of series and parallel tuned LCR circuits.
[6 marks]
In Figure 19, convert the parallel combination of C and R2 to an equivalent series form. Hence show that
the resonant frequency is:
ωo =
1
1
− 2 2
CL C R2
[12 marks]
Discuss briefly how you would determine a value for the Quality factor for this circuit.
[8 marks]
Q23
Discuss briefly the effects of resistive loading on a parallel tuned circuit and show that the effective or
loaded quality factor Q is given by
QUL
QL =
R
1+
Rp
Where Q is the unloaded Q-factor, Rp is the dynamic resistance of the unloaded circuit and R is the
resistive loading in parallel with the circuit.
[8 marks]
Q24
Explain what is meant by the loaded Q-factor of a circuit. A single tuned rf amplifier stage using a self
-biased common source JFET is required to operate at a centre frequency of 1 MHz with a bandwidth of
100 kHz. Calculate the necessary circuit component values for this amplifier if the transistor dynamic
drain resistance is 100kΩ and the transconductance is 2 mS. (It can be assumed that the total parasitic
capacitance at the output have a value of 100 pF). Discuss briefly how the bandwidth of this circuit could
be doubled.
[4 marks]
6
Copyright Paul Tobin, School of Electronics and Communications Engineering
DIT, Kevin St,
Signals and systems
S008_2
Q25
The effective primary impedance of an air-cored transformer when terminated in an impedance Z is
defined as where is the primary impedance and impedance. The parameters of the transformer are defined
s =10 uH. The Q-factor for the primary circuit is 15 and that for the secondary is 50. Calculate the
effective primary impedance at a frequency = 2.5 MHz when the secondary is terminated in. Determine
the resonant frequency and the Q-factor for the circuit.
Q26
Discuss how the source impedance affects the selectivity of series and parallel tuned LCR circuits.
[6marks]
For the circuit shown in figure 19, determine from first principles, an expression for the resonant
frequency and the Q-factor. Hence determine a value for the resonant frequency and the Q-factor for the
values given.
R1 = 100 kΩ, R2 = 40 Ω, L = 100 μH, C = 200 pF.
[14 marks]
Q27
A current source is connected across a parallel tuned LC circuit which is formed by a 1 mH inductance
whose Q- factor is 50 and a loss-free capacitance C. Measurements carried out on this circuit yielded the
following results:
Resonant frequency = 100 kHz
Lower - 3 dB frequency = 98 kHz,
Upper - 3 dB frequency = 102 kHz,
Determine the value of the tuning capacitance C for resonance. What value of source impedance Rs, the
series resistance of the inductor, at the resistance R which, when connected across, will double the circuit
bandwidth, and the value of a resistance which when placed in series with the inductance will produce
the same effect as in above. State any assumptions made in the above analysis.
[14 marks]
Q28
Convert the parallel circuit formed by CR2 in figure 19 to an equivalent series circuit.
[6 marks]
Hence determine the resonant frequency and Q-factor for the complete circuit.
Q29
Explain the concept of resonance in a tuned circuit.
Obtain a relationship for the current in a series tuned circuit normalised to the current at resonance. The
expression should be in terms of the Q-factor and the resonant frequency fo. Calculate the current in a
series tuned circuit at a frequency 10 % off resonance when 10 V pp is applied across the circuit. The
7
Copyright Paul Tobin, School of Electronics and Communications Engineering
DIT, Kevin St,
Signals and systems
S008_2
reactance of the inductance at fo is 200 and the total circuit resistance is 10 Ω. Explain what is meant by
the loaded Q factor of a tuned circuit
[4marks]
Q30
Sketch the form of the impedance and phase variation in a parallel tuned circuit. Explain why it is
normally fed from a current source. A tuned circuit consists of a 100 uH coil and a variable capacitance
connected in parallel. The ac resistance of the coil is 10 Ω. If the circuit is required to be resonant at 1
MHz, calculate the dynamic resistance, bandwidth and capacitance value. What is the value of the
impedance and phase of the circuit 10 kHz off resonance?
Discuss briefly the effects of resistive loading on a parallel tuned circuit and show that the effective or
loaded quality factor Q is given by where Q is the unloaded Q-factor, Rp is the dynamic resistance of the
unloaded circuit and R is the resistive loading in parallel with the circuit.
[8marks]
Q31
Calculate the loaded quality factor for the circuit in figure. The tuned circuit components have the same
values and the coefficient of coupling between the two section of the coil is unity. Justify any
approximations used in the solution.
N1 = 10 T, N2 = 10 T, rds = 10 kΩ.
Q32
The parameters of a parallel tuned circuit are:
Inductance = 470 uH, resonant frequency = 1 MHz, and the - 3 dB bandwidth is 25 kHz. If the total
resistive loading on the circuit is 200 kΩ, calculate:
(a) The loaded Q-factor,
(b) The tuning capacitance C,
(c) The ac resistance of the coil, and
(d) The value of an external resistance which, when added to the above circuit will double the bandwidth.
[12 marks]
Q33
Obtain an expression for the ratio of the impedance of a parallel-tuned circuit off resonance to the
impedance at resonance.
[12 marks]
The impedance of a coil is 5 + j2πf10 Ω. Connected across this impedance is a 10 nF capacitance. Using
the formula derived above, calculate the impedance ratio a decade above the resonant frequency.
Q34
In the circuit shown in Figure 20, the coil has an inductance L = 20 uH and a Q-factor of 25 at ω = 10 rs,
the resonant frequency of the circuit.
a) Find the value of C,
b) Calculate the value of R required to give a loaded Q-factor of 20,
c) Determine the circuit bandwidth, and
d) Calculate the impedance at resonance.
Q35
A parallel tuned circuit consists of a capacitor C in shunt with an inductor L of series ac resistance 10 Ω.
Derive equations for the frequency of resonance and the dynamic impedance of the circuit.
[12 marks]
Determine the frequency of resonance and the dynamic impedance if L= 500 μH and C= 68 pF.
[4 marks]
Q36
Convert the parallel R L section to an equivalent series section in the tuned circuit shown in Figure 21.
[8 marks]
8
Copyright Paul Tobin, School of Electronics and Communications Engineering
DIT, Kevin St,
Signals and systems
S008_2
Derive an equation for the resonant frequency of the circuit and determine its value if R=1000 kΩ,
C=100 nF and L=1000 μH.
[12 marks]
Give an example of an application for a parallel tuned circuit.
[4 marks]
Q37
(c) Replace the parallel RC section in the resonant circuit shown in Figure 22. with an equivalent series
section. Determine the resonant frequency of the circuit.
[5 marks]
Fig.22. Resonant Circuit.
Q38
For the circuit shown in Fig 23, determine from first principles, an expression for the resonant frequency
and the Q- factor. Hence determine a value for the resonant frequency and the Q-factor for the values
given.
R1 = 100 k Ω , R2 = 40 Ω , L = 100 μH, C = 200 pF.
[20 marks]
Q39
A 470uH inductance, whose ac resistance is 25 Ω, has a 5.4 nF connected in series. The circuit is fed
from a voltage source with resistance 50 Ω. Show how the relationship between the loaded Q-factor QL
and the unloaded Q-factor QUL is:
Where Rs is the source resistance and R the ac resistance of the coil.
[12 marks]
With the aid of suitable diagrams, explain how the am receiver could be implemented using digital
signal techniques.
[8 marks]
Q40
A series-tuned LCR circuit as shown in Figure 24, has the output taken across the resistance R .
Transform the circuit using the Laplace operator and hence obtain the voltage transfer function.
[10 marks]
Hence plot the pole - zero constellation and discuss the implications of lowering the value of the
resistance R.
R = 1 Ω , L = 1 H , C = 1 F.
[10 marks]
Mesh and nodal Analysis
Q41
(a) Apply mesh analysis to the circuit shown in Figure 25 and write the resultant equations in matrix
form. Hence solve for the current in R2 using determinant algebra.
[14 marks]
(b) Write a set of equations for the node voltages v1, v2, and v3 expressing the equations in matrix form
(Do not solve for any node voltages).
R1 = 1 Ω, R2 = 2 Ω, Xc1 = -j1 Ω, Xc2 = -j0.25 Ω, Xc3 = -j0.25 Ω, XL1 = j4 Ω, V1 = 5∠60 V.
[6 marks]
9
Copyright Paul Tobin, School of Electronics and Communications Engineering
DIT, Kevin St,
Signals and systems
S008_2
Figure 25
Q42
In figure 26, determine, using mesh analysis and matrix algebra, the current in R1.
[12 marks]
Give the equations in matrix form for the nodal voltages V1 and V2. A solution for these voltages is not
required.
[8 marks]
R1 = 10 Ω, R2 = 20 Ω, R3 = 15 Ω, R4 = 25 Ω, X1 = j10 Ω, X2 = -j5 Ω, X3 = j5 Ω, E= 100/215
Q43
Write the nodal equations for the network in Figure 27 and, solving by determinants, calculate values for
the node voltages and hence the current in C. Determine the resonant frequency and the Q-factor for the
circuit in Fig. 28.
Apply nodal analysis to the circuit diagram in Figure 29 and calculate the voltages at node a and b. Hence
calculate the current in R State two other circuit analysis techniques which could have been used in
determining the current in R.
[20 marks]
Q44
Design from first principles a low pass network using a capacitor and an inductor, suitably connected, to
achieve correct matching between a source Z = (300 +j 0)Ω and a load Z = (600 + j0) Ω if the operating
frequency is 10 MHz
[20 marks]
For X = j5 Ω, R = 10 Ω, R = 2 Ω, R = 3 Ω, X = - j4 Ω, X = - j10 Ω and e = 25 V in Fig. 30, obtain (a) the
nodal equations expressed in matrix form, and
[8 marks]
(b) The voltage values at node 1 and 2.
[12 marks]
R1 = 20 Ω, R2 = 10 Ω, XC1 = - j20 Ω, XC2 = - j10 Ω, Xl = j20 Ω.
Q45
Calculate the current in R1 and R2 for the circuit drawn in figure 29. The mesh equations should be given
in matrix format and the currents solved using determinant algebra.
[20 marks]
10
Copyright Paul Tobin, School of Electronics and Communications Engineering
DIT, Kevin St,
Signals and systems
S008_2
Q46
Figure 30
Q47
The circuit shown in figure 30 is a Hay bridge. Using mesh analysis and solving by determinants, show
that the balance condition (current in RL = 0) obtains for the condition
Q48
In figure 31 determine by nodal analysis and solve by determinants, the node voltage V and hence the
current in R.
[20 marks]
Two-Port networks
Q49
Define image impedance.
[4 marks]
An asymmetrical resistive tee-attenuator has series arms Z1 = 300 Ω, Z2 = 600 Ω and shunt arm Z3
=100 Ω. Derive expressions and hence calculate values for the image impedances Zo1 and Zo2
[8 marks]
Calculate the insertion loss, expressed in nepers, when the network is connected on an image basis.
[8 marks]
Q50
Design from first principles a resistive network that will introduce an attenuation of 9 dB when operating
between a source and load resistance of 50 Ω.
[10 marks]
Design from first principles a low pass network using a capacitor and an inductor, suitably connected, to
achieve correct matching between a source Zs = 300 + j 0 Ω and a load ZL = (600 + j0) Ω if the
operating frequency is 10 MHz.
Q51
Distinguish between the characteristic impedance and the image impedance of a two-port network.
a) Calculate the resistor values of a symmetrical Tee attenuator which, when operating between source
and load resistances of 50 Ω, will introduce an insertion loss of 9 dB.
b) Design from first principles a network which will correctly terminate three 75 resistive loads when fed
from a 75 Ω resistive source. Hence calculate the insertion loss for one load.
Q52
The following open-circuit current and voltage measurements were determined experimentally for an
unknown two-port network:
11
Copyright Paul Tobin, School of Electronics and Communications Engineering
DIT, Kevin St,
Signals and systems
S008_2
V1 = 100 V, V2 = 75 V, I1 = 12. 5 A
with the output current equal to zero.
When the input current was set to zero it resulted in the following results.
V1 = 30 V, V2 = 50 V, I2 = 5 A.
Determine the z parameters for this network and hence draw the z equivalent circuit. Calculate the output
current in a 10 Ω resistive load when an ideal voltage source E = 10sin2π10t volts is applied to the input
terminals. What is the value of the input impedance of the network with this load connected?
Calculate a value for the reverse transfer admittance using the Z parameters obtained in the first part of
the question.
Q53
An R.F transmitter whose output impedance is 300 + j0 Ω required to be connected to an aerial whose
impedance is 150 + j0 Ω. Design from first principles a suitable low-pass LC network which will achieve
correct matching between the transmitter and the aerial at an operating frequency of 10 MHz. Outline the
design procedure for an LC network which would be used where the aerial impedance is greater than the
transmitter output impedance.
[20 marks]
Q54
Design from first principles a low pass network using a capacitor and an inductor, suitably connected, to
achieve correct matching between a source Z = 300 + j0 Ω and a load Z = 600 + j0 Ω if the operating
frequency is 10 MHz.
[20 marks]
For X = j5 Ω, R = 10 Ω, R = 2 Ω, R = 3 Ω, X = - j4 Ω, X = -j10 Ω and e = 25 V in figure 31, obtain the
nodal equations expressed in matrix form.
[8marks]
Q55
Explain the meaning of the following terms as applied to two-port networks:
(i) characteristic impedance
(ii) insertion loss
[6 marks]
Derive an expression for the characteristic impedance of the lattice network shown in Fig. 32.
[6 marks]
The above lattice network has the following parameters:
(i) open circuit impedance = 680 Ω
(ii) short circuit impedance = 529 Ω
(iii) Z1 = 1000 Ω
Determine the circuit characteristic impedance and the value of Z2.
[8 marks]
Q56
Define the term image impedances.
Obtain from first principles an expression for the characteristic impedance of a symmetrical resistive
attenuator. Calculate the component values of a symmetrical T-attenuator operated under matched
conditions for a source whose impedance is 600 Ω if the required attenuation is 9 dB.
Q57
Define the term characteristic impedance
12
Copyright Paul Tobin, School of Electronics and Communications Engineering
DIT, Kevin St,
Signals and systems
S008_2
[4 marks]
A Tee network has series arms Z1/2 ohms and shunt arm Z2 ohms. Show that the characteristic
impedance can be expressed as the square root of the product of the short-circuit impedance and opencircuit impedance.
[6 marks]
Design from first principles a resistive network that will introduce an attenuation of 9 dB when operating
between source and load resistance of 50 Ω.
[10 marks]
Q58
Define the Z-parameters for a two-port network and hence draw a Z - equivalent circuit.
[6 marks]
A two -port network is terminated in an impedance Z. Using z-parameters, obtain expressions for the
current gain and the input impedance.
[14 marks]
Q59
Define image impedance.
[4 marks]
An asymmetrical resistive Tee attenuator has series arms Z1 = 300 Ω, Z2 = 600 Ω and shunt arm Z3 =
100 Ω. Derive expressions and hence calculate values for the image impedances Zo1 and Zo2.
[8 marks]
Calculate the insertion loss, expressed in nepers, when the network is connected on an image basis.
[8 marks]
Q60
Define the term characteristic impedance
[4 marks]
A Tee network has series arms Z1/2 ohms and shunt arm Z2 ohms. Show that the characteristic
impedance can be expressed as the square root of the product of the short-circuit impedance and opencircuit impedance
[6 marks]
Q61
Draw a y-equivalent circuit and hence define the y-parameters for a two-port network.
[6 marks]
Evaluate the y - parameters for the network in figure 33 by considering the network formed by R, and
formed by R, and R to
Q62
A two-port symmetrical Tee network has series arms Z and arm Z. Show that the characteristic
impedance Zo is the square root of the product of the open and short circuit impedances.
Calculate the elements of a resistive tee attenuator that will introduce an insertion loss of 9 dB when
correctly terminated in a load ZL = 300 + j0 Ω. Obtain the Thévenin and Norton equivalent circuits for
the circuit in figure 35. Hence determine the current in an impedance Z for the condition of maximum
power transfer.
Q63
Define the y-parameters of a two-port network and draw a y- equivalent circuit.
[6 marks]
The y-parameters of an RF amplifier measured at a frequency of 10 MHz are:
y = 0. 6 + j1. 5 ms, y = 0. 0003 ms, y = 100 - j20 ms, y = j0. 1 ms.
Calculate the input admittance of the amplifier when terminated in y = j 0.2 mS.
[4 marks]
13
Copyright Paul Tobin, School of Electronics and Communications Engineering
DIT, Kevin St,
Signals and systems
S008_2
Digital Signal Processing
Q64
(a)Design a second-order digital filter, with a centre frequency of 7 kHz, and a Q-factor of 36.
Sketch the pole-zero map in the Z-plane, and sketch the form of the frequency response. The
sampling frequency is 20 kHz.
[10 marks]
(b) Derive an expression for the magnitude response of the linear, time-invariant) LTI) system
shown in Figure 37.Calculate the cut-off frequency in Hz, if a0 = 0.5, and T = 40 us (the sampling
period).
[12 marks]
x(n)
y(n)
Σ
1-a0
z-1
a0
Figure 37
(c) Calculate the pole frequency in Hz, and the pole radius of the second-order system whose
system function is:
H (z) =
1 + 0.5 z
1
−1
+ 0.6 z
−2
The sampling frequency is 10 kHz.
[7 marks]
Q65
Obtain the system function for the network in Figure 38 if a0 = 0.5
[10 marks]
Plot the amplitude response.
x(n)
y(n)
Σ
1-a0
z-1
a0
Figure 38
[10 marks]
Q66
(b)The pole-zero plot for an active filter system is shown in figure 39. Deduce the transfer function.
Note the symbol 'x' represents a pole and the symbol 'o' a zero. All poles and zeroes have been
accounted for.
[8 marks]
(c)
[8 marks]
14
Copyright Paul Tobin, School of Electronics and Communications Engineering
DIT, Kevin St,
Signals and systems
S008_2
Figure 39
Q67
The input analogue signals to the filter are sampled 1.0 kHz. Obtain the system function and hence sketch
the pole/zero plot for this filter
[7 marks]
Determine the magnitude of the system function at 0 Hz and 250 Hz.
[8 marks]
Calculate the values of the impulse response h(n) for n = 0, 1, 2, 3, 4.
[5 marks]
Q68
From the difference equations given, draw the block diagram which represents the input and output
variables in terms of delays units, multiplication factors and adders.
a) y(n) = x(n) - 0.5y(n - 1),
b) y(n) = x(n) + 0.5x(n - 1) - 0.9 y(n - 1)
[10 marks]
Determine the system function for each difference equation. Explain how the pole- zero diagram is used
to calculate a value for the system function at a given frequency.
[10 marks]
Q69
A first-order digital filter is described by the following difference equation:
y(n) = x(n-1)+0.6 y(n-1)
The input analogue signals to the filter are sampled 1.0 kHz. Obtain the system function and hence sketch
the pole/zero plot for this filter
[7 marks]
Determine the magnitude of the system function at 0 Hz and 250 Hz.
[8 marks]
Q70
Obtain the system function H (Z) for the DSP system represented in block diagram form as shown in
Fig.40. Hence give an expression for the system function expressed as H(θ).
[12 marks]
Determine the stability of the system by plotting the pole-zero constellation for this transfer function.
[8 marks]
15
Copyright Paul Tobin, School of Electronics and Communications Engineering
DIT, Kevin St,
Signals and systems
S008_2
Q71
a) Draw a block diagram illustrating the main signal processing in a typical digital signal processor. State
briefly the function of each block
[4 marks]
b) Sketch the following digital signals:
u(n+3)
δ(n-3)
u(n) - u(n-3)
[4 marks]
c) Write the equation which defines y(n) in Fig 41.
[2 marks]
d) The input signal x(n) to a DSP system has 4 finite values at 1, 2, 3, -1 and the impulse response of this
system starts at n= 0
Q72
Determine the stability of the system by plotting the pole-zero constellation for this transfer function.
[8 marks]
Determine the system function for each difference equation. Explain how the pole-zero diagram can be
used to calculate a value for the system function at a given frequency and also to assess the system
stability.
[10 marks]
Q73
a) Describe briefly the technique of digital convolution to obtain an output response sequence y(n) to an
input x(n) when the impulse response is known.
[10marks]
b) From the definition of the unilateral Z transform
Obtain the Z-transform of the step function f(n) = u(n).
Determine the magnitude of the system function at 0 Hz and 250 Hz.
[8 marks]
Calculate the values of the impulse response h(n) for n = 0, 1, 2, 3, 4.
[5 marks]
Q74
Write the difference equation for the system shown in figure 42.
[4 marks]
Find an expression for the sequence y(n) such that
y(n) - 7y(n - 1) + 10y(n - 2) = 0
given that y(-1) = 16 and y(-2) = 5, a = 1.5, b = -3
[16 marks]
Q75
Write the difference equation for the system shown in Fig 43.
[4 marks]
16
Copyright Paul Tobin, School of Electronics and Communications Engineering
DIT, Kevin St,
Signals and systems
S008_2
Figure 43
[4 marks]
Write an equation, which defines the output signal y(n) in Fig.43. Hence determine by convolution a
solution for y(n) given that x(n) is defined as x(n) = u(n) - u(n-3) where u(n) is a step function.
Q76
Discrete convolution of two signals x(n) and h(n) is defined by: where x(n-k) is the input signal to an LTI
system and h(k) is the systems impulse response. Write an equation, which defines the output signal y(n)
in Fig.44. Hence determine by convolution, a solution for y(n)given that x(n) is defined as
x(n) = u(n) – u(n-3) where u(n) is a step function.
Q77
Determine the system function for each difference equation. Explain how the pole- zero diagram can be
used to calculate a value for the system function at a given frequency.
[10 marks]
Q78
Describe the technique of digitally convolving an input x(n) with a known impulse response to produce
an output y(n).Illustrate your answer using two signals defined as:
Q79
Digital filters can be classified according to the equation
m
L
k =0
k =1
y ( n) = ∑ a ( k ) x ( n − k ) − ∑ b( k ) y ( n − k )
Use this definition to classify FIR and IIR type filters.
[4 marks]
Obtain the system function H(Z) for the two systems shown in figure 44a and 44b. State that system is a
FIR type filter and which is a IIR type filter.
[10 marks]
Q80
The transfer function for a digital filter is
Z2 - 0.2Z - 0.08
H (Z ) =
Z - 0.25
a) Describe briefly the technique of digital convolution to obtain an output response sequence y(n) to an
input x(n) when the impulse response is known.
[10 marks]
b) From the definition of the unilateral Z transform obtain the Z- transform of the step function f(n) =
u(n).
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Copyright Paul Tobin, School of Electronics and Communications Engineering
DIT, Kevin St,
Signals and systems
S008_2
∞
f ( z ) = ∑ f (n).Z −n
0
[20 marks]
Q81
Sketch the pole-zero plot for the system shown in figure 45. Hence using a graphical technique, evaluate
the magnitude of the system function at frequencies f = 0, fs/4, fs/2. (fs is the sampling frequency).
[6 marks]
Q82
Discrete convolution of two signals x(n) and h(n) is defined by:
m
y ( n) = ∑ x ( k ) h( n − k )
k =0
Where x(n-k) is the input signal to an LTI system and h(k) is the systems impulse response. The input
signal x(n) to such a system starts at n = 0 and has six finite values at 1, 2, 3, 1, -1, 1. The impulse
response of this system starts at n = 0 and has three finite sample values:
Using the convolution sum as defined, obtain the output response y(n) and verify that the Z transform of
y(n) is the product of the transform of x(n) and h(n).
Q83
Obtain the system function H (Z) for the DSP system represented in block diagram form as shown in
figure 46. Hence give an expression for the system function expressed as H(θ).
[12 marks]
Figure 46
Determine the stability of the system by plotting the pole-zero constellation for this transfer function.
[8 marks]
Q84
∞
The convolution sum of two signals h1(n) and h2(n) as shown in figure 47 is : y (n) = ∑ h1 (n).h2 (n)
n =0
Figure 47
Obtain the output sequence y(n), showing the procedure in table form.
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Copyright Paul Tobin, School of Electronics and Communications Engineering
DIT, Kevin St,
Signals and systems
S008_2
[20 marks]
Q 85
Describe the technique of digitally convolving an input x(n) with a known impulse response to produce
an output y(n). Illustrate your answer using two signals defined as:
0 for n < 0
⎡
⎢
x(n) = ⎢1 for 0 ≤ n ≤ 3 and
⎢⎣
0 for n > 3
⎡
0 for n < 0
h(n) = ⎢⎢
1 for n ≥ 0
⎢⎣
Q86
Obtain the system function H(Z) for the DSP system shown in Figure 48. Hence give an expression for
the system function expressed as H(θ). Plot the pole-zero constellation for this system Use this pole-zero
map to determine the stability of the system. Determine the stability of the system by plotting the polezero constellation for this transfer function.
[8 marks]
Obtain the system function H (Z) for the DSP system shown in Fig 49 .
[8 marks]
Hence give an expression for the system function expressed as H(θ).
[4 marks]
Plot the pole-zero constellation for this system
[4 marks]
Use this pole-zero map to determine the stability of the system.
[4 marks]
Q87
(a) Obtain the system function H(Z) for the DSP system in Figure 50. Re-work the system function and
express it H(θ). Plot the pole-zero constellation and hence write a brief note on the stability of the system.
a = 0.25, b = 1.5, b1 = 2.
(b) From the difference equations below, draw the block diagram, which represents the input and output
variables in terms of delay units, multiplication factors and summing blocks.
(i) y(n) = x(n) - 0.5x(n-1), and
(ii) y(n) = x(n) + 0.5x(n-1) +0.5x(n-2) - 0.9y(n-1)
Hence determine the system function for each difference equation.
[8 marks]
(c)Draw the pole-zero diagram for (i) and use the diagram to explain how the system function may be
evaluated at a particular frequency.
[4 marks]
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Copyright Paul Tobin, School of Electronics and Communications Engineering
DIT, Kevin St,
Signals and systems
S008_2
Figure 50
Q88
(a) Deduce an expression for the instantaneous value of a sampled sinusoidal signal ,whose frequency
is fa and is sampled at a frequency fs (Period Ts).
[8 marks]
(b)The difference equations for a DSP system is:
y ( n ) = x ( n ) + 0.25 y (n − 1) + 2 y ( n − 2)
Obtain the system function and hence plot the pole-zero and comment on the stability.
What is the order of this system?. State the minimum number of delay units required to
implement this function.
[6 marks]
(c)The difference equation for a first-order digital filter is:
y(n) = x(n)+ x(n - 1) + 0.5y(n - 1)
Calculate the values of the impulse response h(n), for n = 0, 1, 2, 3, 4.
Q89
(a) Draw the following DSP signals and define the region over which the signals exist
(i)
A unit step signal, and
(ii)
A unit impulse function
Show how the signal in (ii) may be constructed using the DSP signal in (i). Illustrate your
answer using a block diagram consisting of a summing block and a delay block.
[6 marks]
(b)The pole-zero plot for a system is shown in figure 51. Deduce the system function and hence work
out the difference equation. Note the symbol 'x' represents a pole and the symbol 'o' a zero. All poles
and zeroes have been accounted for.
[8 marks]
(c) Give an expression for the convolution of two digital signals x(n) and h(n). Discuss briefly the
operation of convolution (The flip and slip technique). Illustrate your answer using the two signals in
(i) and (ii)
[8 marks]
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Copyright Paul Tobin, School of Electronics and Communications Engineering
DIT, Kevin St,
Signals and systems
S008_2
Figure 51
Transient questions
Q90
Explain how a simple CR filter may be used to both integrate and differentiate a squarewave. Sketch to a
common time scale the input and output waveforms in both cases. Show how the 10% to 90% rise-time
for this network connected as a low pass filter is given as:
τ = 2.2CR
Explain how the upper and lower –3 dB cut-off frequencies of an amplifier can be quickly assessed by
application of a suitable squarewave to the input terminals.
Q91
With reference to the circuit in Fig 52, Sw1 is closed at time t = 0 seconds and the capacitor C was
initially uncharged. Obtain an expression, using the Laplace Transform, for the current in C as a function
of s. (hint: apply Thévenin's theorem to the left of C treating C as the load)
[8 marks]
Hence obtain an expression for the voltage across C as a function of time and evaluate this voltage at
time t = 5 seconds.
E = 10 v, R1 = R2 = 10 Ω, C = 1 F
Q92
The unidirectional current waveform in Fig.53a is applied to an ideal 2 mH inductor. Sketch to scale the
voltage waveform across the inductor.
[10 marks]
21
Copyright Paul Tobin, School of Electronics and Communications Engineering
DIT, Kevin St,
Signals and systems
S008_2
Calculate the capacitor voltage 1 s. Sketch to scale the voltage across C.
[10 marks]
Q93
(a) Show how you may transform, from the time domain to the complex frequency plane (The s-plane), the
following electronic components
(i) A resistance R,
(ii) A capacitor C, and
(iii)An inductance, L
Show how initial conditions are represented in (ii) and (iii)
[7 marks]
(b) An series-tuned LCR circuit, fed from an ideal voltage sine source Vin(s), has the output voltage
Vout(s), developed across the resistance. Obtain the transfer function H(s) expressed as:
V ( s)
H ( s ) = out
Vin ( s )
[7 marks]
(c) For the transfer function in part (b), determine the poles and zeros and hence plot the pole-zero map
for the circuit values given. What is the effect on the poles and hence the response of the circuit, by
halving the value of the resistance in the circuit.
R =1 Ω, L = 1 H, C = 1 F.
[6 marks]
Q94
Transform the circuit elements in Fig 53 using the Laplace transform. Hence obtain a value for the
current in R1 when the switch is closed at time t = 1 second.(assume all circuit elements are initially
uncharged). Hint: Apply mesh analysis and solve using determinant algebra.
R1 = 1 Ω, R2 = 0.5 Ω, R3 = 1 Ω, L = 2 H. B = 2 v
[20 marks]
Q95
Show that a capacitor charged to a voltage V may be represented as a capacitive reactance 1/sC in series
with a voltage source V/s(where V is the initial charge on the C).
` [6 marks]
A 10 H coil with a measured resistance of 1600 Ω (is connected across the terminals of a capacitor which
had been previously charged to 100 V. Obtain an expression for the voltage VL(t) across the coil after the
capacitor has been connected. Calculate a value for this voltage at t = 1 second.
[14 marks]
Q96
A 1-volt step is applied to this circuit. Sketch to scale the output voltage over the period 0 to 5 seconds
for the following circuit values:
R1 = 10 kΩ, R2 = 100 kΩ, C = 10 nF.
[12 marks]
Q97
In Fig.54, the switch is closed at t = 0 s. (assume C1 is uncharged). Obtain an expression for the voltage
across the capacitor Vc(s) and hence, using the Laplace tables supplied, calculate a value for the voltage
Vc(t) at a time t = 1.0 s.
[20 marks]
Q98
Obtain a Thévenin equivalent circuit for the circuit shown in Fig 55, if the load is formed by R3 and L.
[12 marks]
22
Copyright Paul Tobin, School of Electronics and Communications Engineering
DIT, Kevin St,
Signals and systems
S008_2
Calculate a value for the current in R3 if the value for L is chosen as the complex conjugate of the
Thévenin impedance. You may assume the voltage source is ideal.
E = 50∠50 V, R1 = R2 = R3 = 5 Ω, XC = 2 Ω,
[8 marks]
Q99
In Fig.56, the switch is initially opened. The switch is then closed 0.1 seconds after the application of the
1 V step. Sketch to scale the current waveform i(t) over the period 0 <t < 0.5 sec.
E = 1 V, R1 = R2 = 5 Ω, L = 2 H.
[20 marks]
Q100
(a)Sw1 in Figure 57, is closed at time t = 0 seconds where the inductance L1, was initially uncharged.
Obtain an expression, using the Laplace Transform, for the current in L1 as a function of the complex
frequency variable s. (hint: Apply Thévenin's theorem to the left of L1, treating L1 and R3 as the load).
Hence evaluate the current at t = 0.2 s. Make a brief sketch of the current flowing in the circuit.
[14 marks]
(b)Obtain an expression for the voltage across R3 as a function of time and evaluate this voltage at time t =
0.2 seconds.
V1 = 2 v, R1 = R2 = 2 Ω, R3 = 1 Ω, L1 = 1H.
[6 marks]
Figure 57
23
Copyright Paul Tobin, School of Electronics and Communications Engineering
DIT, Kevin St,
Signals and systems
S008_2
Q101
For the circuit shown in figure 58 calculate, by developing an appropriate Thévenin equivalent circuit,
the value of R which will be required to achieve maximum power transfer into the resistor R.
[20 marks]
For X = j5 Ω, R = 10 Ω, R = 2 Ω, R = 3 Ω, X = - j4 Ω, X = - j10 Ω and e = 25 V in figure 1, obtain
(a) The nodal equations expressed in matrix form, and
[8marks]
Q102
Design from first principles, a low - pass network using a capacitor and an inductor suitably connected,
to achieve correct matching between a 300 Ω resistive source and a 600 Ω resistive load if the operating
frequency is 10 MHz.
Calculate the node voltages at 1 and 2 in figure 2 using the input and transfer admittances.
Determine the resonant frequency and the Q-factor for the circuit in figure 58.
Q103
(a) A first-order digital filter is described by the system function
1 + 0.5z −1
H ( z) = k
1 − 0.5 z −1
Let k = 1/2.
(i) Determine k, so that the maximum value of |H(θ)| is equal to 1.
(ii) Compute the 3-dB bandwidth of the filter H(z), and
(iii) Show how this function could be realised in block diagram form using delay, multipliers
and summer units.
(b) Determine the transfer function of the system shown in Figure 59. Check the stability of the system
when r = 0.9 and θ0 = π/4.
1. A digital system is shown in Figure 1.
y(n )=A s in(n+ 1)θo
x(n)
+
z -1
-b1
z -1
-b2
Figure 59
Q104
(a) Assuming θo is the resonant frequency of the digital oscillator, find the values of b1 and b2 for
sustaining the oscillation.
(b) Write the difference equation for Figure 1.Assuming x(n) = (Asinθo)δ(n), and y(-1) = y(-2) = 0,
show, by analysing the difference equation, that the application of an impulse at n = 0 serves the purpose
of beginning the sinusoidal oscillation, and prove that the oscillation is self-sustaining thereafter.
(c) By setting the input to zero, and under certain initial conditions, sinusoidal oscillation can be obtained
using the structure in Figure 1. Find these initial conditions.
(d) Using the values obtained for b1 and b2 in (a) above, show that
24
Copyright Paul Tobin, School of Electronics and Communications Engineering
DIT, Kevin St,
Signals and systems
S008_2
fs Δb
1
⎛ 2π fo ⎞
⎟
4π sin ⎜
⎝ fs ⎠
where fs= sampling frequency (16 kHz) and = desired frequency of oscillation. Show that the highest
frequency resolution Δfo that is obtainable for this oscillator is 0.078 Hz. You may assume that b1 is
1
represented by a K-bit number (fractional arithmetic) and Δb1 is given by Δb1 =
where K = 16.
2
K
−
2
Δfo =
Q105
Derive an expression for the magnitude frequency response of the second-order discrete-time system
with the following transfer function:
H ( z) =
1
1 + 1.3435z
−1
+ 0.9025z
−2
What is the resonant frequency (in Hz) of the system, if the sampling frequency is 20 kHz?
[10 marks]
Q106
Explain the terms minimum-phase function and non-minimum phase function. Show how the network in
figure 2 is a non-minimum phase network and state one application for this network.
A system with left -hand zeros only are called minimum - phase functions. It can be seen from a pole zero plot of such functions that the angle of the transfer function will vary over a much angle than a
system, which has right -hand zeros. Considering the all-pass network in figure 2 the angle for the
transfer function H(s) is given as:
One application for this network is where a signal is suffering from an unwanted phase lag then the
network can supply enough additional phase lag to bring the total angle to 0.
Q107
Obtain the voltage transfer function H(s) and plot the pole-zero constellation for H(s). Hence using this
diagram, show that the network in figure 2 is an ALL-PASS filter
[12 marks]
Q108
A two port symmetrical Tee network has series arms Z an arm Z. Show that the characteristic impedance
Z is the square root of the product of the open and short circuit impedances. Calculate the values of the
elements of a resistive tee attenuator which will introduce an insertion loss of 9 dB when correctly
terminated in a load Z = 300 +j 0
The effective primary impedance of an air-cored transformer when terminated in an impedance Z is
defined as where Zs the primary impedance and 1 MΩ impedance. The parameters of the transformer are
defined as: = 100 uH Ls = 10 uH. The Q-factor for the primary circuit is 15 and that for the secondary is
50. Calculate the effective primary impedance at a frequency = 2.5 MHz when the secondary is
terminated.
Q109
The unidirectional voltage waveform in Figure 1a is applied to a loss-free 10 uF capacitor. Sketch to
scale the current waveform across the capacitor
[10 marks]
25
Copyright Paul Tobin, School of Electronics and Communications Engineering
DIT, Kevin St,
Signals and systems
S008_2
Fig.1b, the switch is moved to position 1 at t = 0 seconds. After one time constant, the switch is moved to
position. Sketch to scale, over a time interval 0 to 2 ms, the complete (E current waveform. assume at
time t = 0 the capacitor is uncharged.
Obtain the system function H (Z) for the DSP system represented in block diagram form as shown in Fig.5.
Hence give an expression for the system function expressed as H(θ).
[ 12 marks]
Q110
Determine the stability of the system by plotting the pole-zero constellation for this transfer function.
[ 8 marks]
Q111
Obtain a Thévenin equivalent circuit for the circuit shown in Fig.1, if the load is formed by R3 and L.
[12 marks]
Calculate a value for the current in R3 if the value for L is chosen as the complex conjugate of the Thévenin
impedance. You may assume the voltage source is ideal.
E= 50 50 R1 = R2 = R3 = 5 Ω, Xc = 2 Ω,
8 marks]
Q112
With reference to the circuit in Fig 5, Sw1 is closed at time t = o seconds and the capacitor C is
initially uncharged. Obtain an expression, using the Laplace Transform, for the current in C as a
function of s. (hint: apply Thévenin's theorem to the left of C treating C as the load)
[8 marks]
Hence obtain an expression for the voltage across C as a function of time and evaluate this voltage at
time t = 5 seconds.
E = 10 v, R1 = R2 = 10 Ω, C = 1 F
[12 marks]
Q113
Convert the parallel R L section to an equivalent series section in the tuned circuit shown in Fig.4.
[8 marks]
Derive an equation for the resonant frequency of the circuit and determine its value if R=1000 kΩ, C=100
nF and L=1000 μH.
[12 marks]
Q114
For the circuit shown in Fig 1, determine from first principles, an expression
for the resonant frequency and the Q- factor. Hence determine a value for
the resonant frequency and the Q-factor for the values given.
R1 = 100 k Ω , R2 = 40 Ω , L = 100 μH, C = 200 pF.
[20 marks]
Q115
A parallel tuned circuit consists of a capacitor C in shunt with an inductor L of series ac resistance 10 Ω.
Derive equations for the frequency of resonance and the dynamic impedance of the circuit.
[12 marks]
Determine the frequency of resonance and the dynamic impedance if L= 500 μH and C= 68 pF.
[4 marks]
Give an example of an application for a parallel tuned circuit.
[4 marks]
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Copyright Paul Tobin, School of Electronics and Communications Engineering
DIT, Kevin St,
Signals and systems
S008_2
Q116
a)
A 470uH inductance, whose ac resistance is 25 Ω, has a 5.4 nF connected in series. The circuit is fed
from a voltage source with resistance 50 Ω. Show how the relationship between the loaded Q-factor
QL and the unloaded Q-factor QUL is:
Where Rs is the source resistance and R the ac resistance of the coil.
[12 marks]
With the aid of suitable diagrams, explain how the am receiver could be implemented using digital
signal techniques.
[8 marks]
Q117
A series-tuned LCR circuit as shown in Fig 3, has the output taken across the resistance R . Transform
the circuit using the Laplace operator and hence obtain the voltage transfer function .
[10 marks]
Hence plot the pole - zero constellation and discuss the implications of lowering the value of the
resistance R.
R= 1 Ω , L = 1 H , C = 1 F.
[10 marks]
Q118
Obtain the transfer function (vo/es) for the circuit shown in Fig.2 and determine the two break frequencies.
[12 marks]
Sketch the circuit asymptotic response. What type of frequency response does the circuit exhibit.
[8 marks]
R1=R2=10 kΩ, C=100 n F.
Q119
(a) Draw the circuit diagram and derive the transfer functions equations of the following first order RC
based filters:
(i) low-pass
(ii) high-pass
(iii) modified low pass
Sketch the asymptotic response of each circuit. Discuss briefly the significance of each response. Explain
the term roll-off.
[9 marks]
(b) In (iii) above, determine the values of the resistors in the circuit if the two break frequencies are 10 kHz
and 100kHz and the capacitance used is 10 nF.
[6 marks]
Q120
(c) Replace the parallel RC section in the circuit shown in Fig.1. with an equivalent series section. Whence
find the resonant frequency of the circuit.
[5 marks]
Obtain the voltage transfer function for the network shown in Fig 5.
[10 marks]
State the values of the break frequencies and hence plot the asymptotic amplitude response.
R1 = 2.2 kΩ , R2 = 470 Ω , C= 4700 pF.
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Copyright Paul Tobin, School of Electronics and Communications Engineering
DIT, Kevin St,
Signals and systems
S008_2
[10 marks]
(b)The pole-zero plot for an active filter system is shown in figure 4. Deduce the transfer function.
Note the symbol 'x' represents a pole and the symbol 'o' a zero. All poles and zeroes have been
accounted for.
[8 marks]
Figure 4
Determine the stability of the system by plotting the pole-zero constellation for this transfer function.
[8 marks]
Obtain the system function H (Z) for the DSP system shown in Fig 4 .
[8 marks]
Hence give an expression for the system function expressed as H(θ).
[4 marks]
Plot the pole-zero constellation for this system
[4 marks]
Use this pole-zero map to determine the stability of the system.
[4 marks]
Fig 4
Q121
(a) A first-order digital filter is described by the system function
1 + 0.5 z −1
H ( z) = k
1 − 0.5 z −1
Let k = 1/2.
(i) Determine k, so that the maximum value of |H(θ)| is equal to 1.
(ii) Compute the 3-dB bandwidth of the filter H(z).
(iii)Show how this function could be realised in block diagram form using delay, multipliers and
summer units.
(b) Determine the transfer function of the system shown in Figure 2. Check the stability of the system
when r = 0.9 and θ0 = π/4.
28
Copyright Paul Tobin, School of Electronics and Communications Engineering
DIT, Kevin St,
Signals and systems
S008_2
1. A digital system is shown in Figure 1.
y(n )=A s in (n+ 1)θo
x(n )
+
z -1
-b1
z -1
-b2
Figure 1
(a) Assuming θo is the resonant frequency of the digital oscillator, find the values of b1 and b2 for
sustaining the oscillation.
(b) Write the difference equation for Figure 1.Assuming x(n) =(Asinθo)δ(n), and y(-1) = y(-2) = 0, show,
by analysing the difference equation, that the application of an impulse at n = 0 serves the purpose of
beginning the sinusoidal oscillation, and prove that the oscillation is self-sustaining thereafter.
(c) By setting the input to zero, and under certain initial conditions, sinusoidal oscillation can be obtained
using the structure in Figure 1. Find these initial conditions.
(d) Using the values obtained for b1 and b2 in (a) above, show that
Δfo =
fs Δb1
⎛ 2π fo ⎞
⎟
4π sin ⎜
⎝ fs ⎠
where fs = sampling frequency (16 kHz) and = desired frequency of oscillation. Show that the highest
frequency resolution (Δfo) that is obtainable for this oscillator is 0.078 Hz. You may assume that b1 is
represented by a K-bit number (fractional arithmetic) and Δb1 is given by
1
Δb1 =
where K = 16.
K
2 −2
(b) Derive an expression for the magnitude frequency response of the second-order discrete-time system
with the following transfer function:
H ( z) =
1
1 + 1.3435z
−1
+ 0.9025z
−2
What is the resonant frequency (in Hz) of the system, if the sampling frequency is 20 kHz?
[10 marks]
29
Copyright Paul Tobin, School of Electronics and Communications Engineering
DIT, Kevin St,
(b)
Signals and systems
S008_2
Design a second-order digital filter, with a centre frequency of 2.7 kHz, and a Q-factor of 1.6.
Sketch the pole-zero map in the Z-plane, and make a rough sketch of the frequency response. The
sampling frequency is 10 kHz.
[10 marks]
(b) Derive an expression for the magnitude response of the linear, time-invariant system shown in Fig3.
If a0 = 0.5, and T = 40 μs (the sampling period), calculate the cut-off frequency in Hz.
[12 marks]
x(n)
y(n)
Σ
1-a0
z-1
a0
Fig. 3
(c) Calculate the pole frequency in Hz, and the pole radius of the second-order system (with a single
conjugate pole-pair) whose transfer function is as follows:
H ( z) =
1 + 0.51298 z
1
−1
+ 0.6489 z
−2
The sampling frequency is 10 kHz.
[7 marks]
x(n)
y(n)
Σ
1-a0
z-1
a0
Fig. 4
The input analogue signals to the filter are sampled 1.0 kHz. Obtain the system function and hence sketch
the pole/zero plot for this filter
[7 marks]
Determine the magnitude of the system function at 0 Hz and 250 Hz.
[8 marks]
Calculate the values of the impulse response h(n) for n = 0, 1, 2, 3, 4.
[5 marks]
:
Fig.1. Resonant Circuit.
(b)
Show how this circuit can be modified using an additional operational amplifier to produce an astable
oscillator. Suggest necessary additional passive component values for this circuit if the output frequency of
the oscillator is 7500 Hz. (Assume V=± 12 V, R1=10 kΩ and R2=30 kΩ)
[10 marks]
Explain the meaning of the following terms as applied to two-port networks:
30
Copyright Paul Tobin, School of Electronics and Communications Engineering
DIT, Kevin St,
Signals and systems
S008_2
(i) characteristic impedance
(ii) insertion loss
[6 marks]
Derive an expression for the characteristic impedance of the lattice network shown in Fig. 3.
[6 marks]
The above lattice network has the following parameters:
(i) open circuit impedance = 680 Ω
(ii) short circuit impedance = 529 Ω
(iii) Z1 = 1000 Ω
Determine the circuit characteristic impedance and the value of Z2.
[8 marks]
Obtain a Thevenin equivalent circuit for the circuit shown in Fig.1, if the load is formed by R3 and L.
[12 marks]
Calculate a value for the current in R3 if the value for L is chosen as the complex conjugate of the Thevenin
impedance. You may assume the voltage source is ideal.
[8 marks]
E= 50 50 R1 = R2 = R3 = 5 Ω, Xc = 2 Ω,
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Copyright Paul Tobin, School of Electronics and Communications Engineering