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Digital Signal Processing ELE8059

创建日期：2024-05-06 17:30:16

所属分类：电气工程electric engineering

标签： ELE8059

Digital Signal Processing

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Digital Signal Processing ELE8059

Q1. Consider a continuous-time system in which the relationship between input
(x) and output (y) is:
( ) ( ) 0.8 1 2a
dy dx
y t a x
dt dt
+ = − + ,
where time is denoted by t and a is a non-negative integer constant.

(a) For which choices for a is this a linear system? Explain your answer (you can
use maths, but a full formal proof of linearity is not required).

[6 Marks]
(b) Specify the transfer function and the zeros and poles of the system for the
case a = 1.
[6 Marks]
(c) For a = 1, is the system stable? Explain your answer. [4 marks]
(d) Again taking a = 1, derive a closed-form expression of the output signal, as
a function of time, when the input signal is a pulse of width T, as plotted in
Figure Q1.

Figure Q1.

[9 Marks]

Digital Signal Processing ELE8059

Q2. A digital infinite impulse response (IIR) high-pass filter is required. The
digital filter is to be designed to operate at a sample rate of 12 kHz with a
cut-off frequency at 3 kHz. One requirement of the filter is that its
amplitude response should have no ripples in the passband. It is decided
that the digital filter design is to be based on the following low-pass
prototype filter transfer function:
3 2
1
( )
2 2 1
H s
s s s
=
+ + +

(a) Explain why the above transfer function is a suitable prototype low-
pass filter choice in this case. [5 Marks]
(b) Use the bilinear transform to derive, from the above prototype transfer
function, an IIR high-pass that has the correct cut-off frequency. Show that
the resulting digital filter transfer function can be written as
( )
31
2
1
( )
1
g z
H z
az
−
−
−
=
+
,
And specify the values of the constants and g a . [12 Marks]
(c) Draw a signal diagram of a “parallel structure” realisation of the digital
filter, featuring a biquad section in parallel with a first-order section.
Specify all coefficients used in the realisation. [8 Marks]

Q3. A signal containing two frequency components (1812 Hz and 2352 Hz) of
equal amplitude is sampled with a digital processor. The sampling
frequency (fs) of the processor can be set to any integer multiple of 100 Hz,
up to 10000 Hz. The signal block length (N) can be set to any integer value
ranging from 200 to 2000 samples. The default settings of the processor
are fs = 4800 Hz and N = 200 samples.
(a) The processor is used to take the Discrete Fourier Transform (DFT) of
an N-sample block of the signal, using the default settings. Sketch the
resulting amplitude response, using bin number on the horizontal axis,
and covering a frequency range from 0 Hz to 2500 Hz. [6 Marks]
(b) Describe the main non-ideality in the amplitude spectrum. Explain
how it can be eliminated, keeping any increase in processing and memory
requirements as low as possible. [8 Marks]
(c) An alternative sampling strategy is considered, using bandpass
sampling. What is the lowest sampling frequency available on the
processor that can be used as such for the given input signal? [11 Marks]

Q4. As indicated in Figure Q4, a digital signal source y(n) consists of a periodic
signal x(n) and additive white noise v(n) of variance 2 1 6v = . One period
of the periodic signal x(n) is given by { 1, 0, 0, -1, 0 0 } .
Figure Q4.

(a) Construct the 6 6 autocorrelation matrix of the additive white noise
signal v(n). Specify the values of the matrix elements. [5 Marks]
(b) Derive the 6 6 autocorrelation matrix of the signal y(n). Specify the
values of the matrix elements. [9 Marks]
(c) Taking the signal y(n) as an input, it is desired to determine a 6-tap FIR
filter which gives a triangular output signal t(n), with one period given
by { -3, -1, 1, 3, 1, -1 } . Specify the coefficients of the optimal 6-tap FIR
filter. [11 Marks]

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FE 825 IERG 5130 STAT 8178 EE161 ADS2 G54FUZ CS326 113B ECN 226 QBUS6860 ELEC3202 COMP9334 MGT11B MET CS 777 STA238 00099F Python SMM306 CMT206 EC116 SP MAST90084 CS480/680 BCPM0091 CMPT 300 ECON2209 FY20 STAT 432 CVEN9625 MATH1712 CMP9771M CMP3750M CIS 325 AI506 MAT344H1S MGT7180 PSTAT 160B CSYS5010 1D STAT 453/558 ME152B STA304H1F CSCI-128 ME428 CO3002 FINE 7640 COMP3234 ECSE 223 EE497 – 3D MATH5905 ECS648U ENG5009 MATH10242 EC665 A CPI221 ISOM5220 MACE40402 MGT B456F COMP9315 AFM 113 STAT 778 CS 564 MA231 CS 230 CS 124 COMP5511 CSC317 ITEC1010 ELEC207 INFT 3019 7CCMFM06 EE 161 5G ECLT5810 MATH 7349 CS526 O2 COMP90054 ELE00063M ACCT5034 ARC1015 CSCI 40 CSCI E40 FIN 524B CS 462 389COM CMT309 MFIN703 COMP4250 STA304 STA304H1S STA 1003 FIN 448 BUFN 732 CS520 CVEN3701 ECMT6006 PHAS0100 CSIT314 MDIA 5028 CSCI 2134 MAT 451 CMPT 459 STAT 457 INFO20003 COMP4141 IELE8066 F540 ECM9 CSE 6321 FIN 430 MATH3911 BFC5916 COMP1011 EEE 6209 STATS 240 CW3 COMP206 COMP1003 4331W ISE 530 CMPT 295 MATH 239 QF5203 INFR08010 IT114105 FIT2093 ECON3124 ECON1195 MATH 337 STAT 3006 COMP202 COMP3130 MATH3609 COMP 250 CS6140 IEEE ELE00041I FIN40090 CIV 5899 MATH3066 IGNMENT MATH 2009 STAT3405 HIST 101 MATH 3281 CEGE0042 ICT532 LAB 7 LAB 5 MCO455 CSIT214 Q4 IFB104 A1 ECO220Y5Y DA5020 CMPSC473 FIN9007 DSCI553 MSCI534 COMP3308 CSE-112 CMPT 310 EE104 F 2C P20 ALY2010 BUSM060 STA258 ECO00030M R BIOSTAT 274 COS6012-B FIT3155 CMPUT 340 EC421 INFS4205 ECOM135 NATS 1740 COMP809 MBE 6005 STA465 CSSE2310 CIVL2700 DATA4700 CMPSC 473 ISE 529 CS5044 MAST30020 COMP2207 ENGG1001 COMP0046 FIT3165 STAT GR5241 MECH0007 vATHK1001 ANALYTIC ATHK1001 MATH352001 HW3 S1 BMAN 24662 COMP 636 Stat-461 STAT 302 STAT 2132 STAR534 ENG4137 CMP3752M CS3103 BISM3222 STA 135 ME155A IIMT3601 ECON3371 ECON 3371 HW4 FINA 5840 DT211C CTEC3902 MAT005 MATH2148 S261F Econ 332 MATH3734 MIS41270 7CCMFM13 SYST 664 CIS434 CENG10014 MT4111 MT4527 ECMM424 CMT219 Math 136 MATH 3120 COMP10002 MAST20009 MATH0058 BEEM117 ISYE 6420 MGMTMFE 405 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CS 333 CP1406 KIT308 Economics 701 ECON3740 ECE 5759 BFC3340 FINC2012 COMP4500 MATH3871 FIT3080 18-819C Economic ECO202Y5Y PHY100 FINC-780 Econ 100B EC602 IE582 EE450 COMP9517 WRDS 150B ECON 570 LECTURE 2 ECA5101 ECE 4420 ADAD9311 FIT3031 CS 544 MF 703 CS 6140 KIT501 ENGSCI 344 CSC171 IE 332 FINM8007 21S2 COSC 1114 FE545 E471 SIT102 CSC108 MAT301 IAE 101 COP3503 STAT2004 EC 303 CS1026 STAT1203 MATH3171 ISOM5160 IFN657 FIT2014 ACCT2102 DSCI550 STATS 330 ELEC421 COMP3670 QBUS6840 MEES:698B MAST30027 MIET1077 Economics IFT 6758 FIT5122 BUSN5101 CptS 111 HRMT5502 Math2018 PP 312 INFS3200 CZ3005 CSSE1001 SENG 2011 MKTG 553Q BUFN 640 COMP507 ETF5952 DATA 311 ECE4043 BANK3011 finance M3H623544 HW1 INFS 7410 AI6103 STAT3888 INFS3208 ACCT7103 ECOS3013 PSYCH 20B F79MA STAT3006 ENGG952 2X MGSC 7000 ECE4132 MSF 526 MATH 3033 CDS 511 EEEE4120 EEET2465 ECA5103 MATH2070 ELEC3104 BU51019 EEE8147 ENGR20003 COSC 1107 ETF3200 FNCE 435 MSFI 435 ISyE6420 GENG4405 HW ETC5523 IE 7315 MOEC0021 STA 106 ENGG 1300A ENGG 1300 OLET1139 STAT 3503 STAT 206 FEM11149 ELEN 4720 COMP20003 SOLA1070 CS574 C11CS LINB29H3 MAST90083 ECON7200 STAT 610 LINB29 DSM-5 COMP3161 EEEN40010 MATH 4431 K353 STA 3386 IERG4300 COMS W4167 ENG5298 AMME 2302 CS3910 COMPSCI 320 STATS 369 GR5242 COSC 328 CSCI435 MGTS1301 INFS2200 COMP3900 CIS 375 Module AE03 COMP3121 CS260 HAM503 MGMT5050 INT3075 COSC 285 GEOL0029 GEGE2X01 STAT1201 MATH1061 MMF1941H STAT 5060 CS 520 COMP 4320 MSIN0104 000T FINS3648 ECMT2160 ECON4003 Phys 129L CSE 142 CS 3130 MGEB01 EECS 340 Finance (20443/20444) ECMM149 COMP5310 EEEN60180 CSC 111 AT2 CSCI 4430 STAB57 MA/CSC 427 MATH322 D100 CSCI544 CITS3004 CI6227 CSE 565 APCO 1P01 852L1 ELEC S411F COMP3207 STATS330 ECON7530 FIT5149 CSI4104 CMPT 371 COMP201 SEHS4696 FI4003 MATH6182 CG1 WS MATH401 MGT 205 QTM 110 COMP 0137 MACE12201 FIT5210 ECE 544 BISM7206 HUDM 6055 ECON3203 STATS 210 MSCI 570 MATH6173 COMPSCI4039 Accounting COM3503 AOE2024 ECON213 PSYC10004 Control Data 100/200A PHIL2617 PHYS3116 MIE250 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