Technical Information Oil and Gas Sector Significant Stationary Sources of NOx Emissions Final
Monitoring Systems at Stationary Sources - US Environmental
Transcript of Monitoring Systems at Stationary Sources - US Environmental
![Page 1: Monitoring Systems at Stationary Sources - US Environmental](https://reader036.fdocuments.us/reader036/viewer/2022071601/613d47ab736caf36b75b7469/html5/thumbnails/1.jpg)
COSC 3451: Signals and Systems
Course Instructor: Amir Asif Teaching Assistant: TBA
Contact Information: Instructor: Teaching Assistant:Office: CSB 3028 Durwas, [email protected] [email protected](416) 736-2100 X70128
URL: http://www.cs.yorku.ca/course/3451
Text: A. V. Oppenheim and A. S. Willsky with S. H. Nawab, Signals andSystems, NY: Prentice Hall, 1997.
Class Schedule: TR 13:00 − 14:30 (CB 115)
Assessment: Assignment / Quiz: 20% Projects: 15%Mid-term: 25%Final: 40%
Office Hours: Instructor: CSB 3028, TR 12:00 – 13:00 TA: TBA
![Page 2: Monitoring Systems at Stationary Sources - US Environmental](https://reader036.fdocuments.us/reader036/viewer/2022071601/613d47ab736caf36b75b7469/html5/thumbnails/2.jpg)
2
Course Objectives
Introduce CT and DT signals and systems in terms of physical phenomena
Review elementary signals and see how these can be represented in domains other than time
Understand different transforms (Fourier, Laplace, Z) used to analyze signals
Complete analysis of a linear time invariant (LTI) systems in time and other domains
Design filters (systems) to process a signal for different applications
Present state-of-art technologies used in communication systems
Use computers for digital signal processing
Demystify terminology !!!!!!
![Page 3: Monitoring Systems at Stationary Sources - US Environmental](https://reader036.fdocuments.us/reader036/viewer/2022071601/613d47ab736caf36b75b7469/html5/thumbnails/3.jpg)
3
What is a Signal?
Signal is a waveform that contains information
System is a model for physical phenomena that generates, processes, or receive signals.
Speech Waveformx(t): Continuous Time
![Page 4: Monitoring Systems at Stationary Sources - US Environmental](https://reader036.fdocuments.us/reader036/viewer/2022071601/613d47ab736caf36b75b7469/html5/thumbnails/4.jpg)
4
Examples of Signals
ImageI[m,n]: 2D DT signal
(a) 1D CT signal x(t)(b) 1D DT signal x[n]
Activity 1: For each of the representations: (a) z[m,n,k] (b) I(x,y,z,t), establish if the signal is CT or DT. Specify the independent and dependent variables. Also, think of an example from the real world that will have the same mathematical representation.
![Page 5: Monitoring Systems at Stationary Sources - US Environmental](https://reader036.fdocuments.us/reader036/viewer/2022071601/613d47ab736caf36b75b7469/html5/thumbnails/5.jpg)
5
Power vs. Energy Signals (1)
Energy:
Power:
Activity 2:a. Consider the sinusoidal signal x(t) = cos(0.5πt). Choosing T = 4, determine the average power of x(t).b. Consider the signal x(t) = 5 sin(2πt) for the interval −1 <= t <= 1 and is 0 elsewhere. Calculate the energy
of x(t).c. Calculate the energy of the signal x[n] = (0.8)n for n >= 0 and is 0 elsewhere.
⎪⎪⎩
⎪⎪⎨
⎧
=
∑
∫∞
∞−
∞
∞−∞
Signals DTfor |][|
Signals CTfor |)(|
2
2
nx
dttxE
⎪⎪⎩
⎪⎪⎨
⎧
+∞→
∞→=
∑
∫
−
−∞
Signals DTfor |][|12
1lim
Signals CTfor |)(|21lim
2
2
N
N
T
T
nxNN
dttxTT
P
![Page 6: Monitoring Systems at Stationary Sources - US Environmental](https://reader036.fdocuments.us/reader036/viewer/2022071601/613d47ab736caf36b75b7469/html5/thumbnails/6.jpg)
6
Power vs. Energy Signals (2)
1. Energy Signals: have finite total energy for the entire duration of the signal. As a consequence, total power in an energy signal is 0.
2. Power Signals: have non-zero power over the entire duration of the signal. As a consequence, the total energy in a power signal is infinite.
Activity 3:Classify the signals defined in Activity 2 as a power or an energy signal.
![Page 7: Monitoring Systems at Stationary Sources - US Environmental](https://reader036.fdocuments.us/reader036/viewer/2022071601/613d47ab736caf36b75b7469/html5/thumbnails/7.jpg)
7
Linear Transformations
There are two types of linear transformations that we will consider
1. Time Shifting:
2. Scaling:
)()( ottxty −=
)()( atxty =
![Page 8: Monitoring Systems at Stationary Sources - US Environmental](https://reader036.fdocuments.us/reader036/viewer/2022071601/613d47ab736caf36b75b7469/html5/thumbnails/8.jpg)
8
Linear Transformations (2)
Scaling for DT Signals:
Note that y[n] = x[n/2] is not completely defined
][][ onnxny −=
![Page 9: Monitoring Systems at Stationary Sources - US Environmental](https://reader036.fdocuments.us/reader036/viewer/2022071601/613d47ab736caf36b75b7469/html5/thumbnails/9.jpg)
9
Linear Transformations (3)
Precedence Rule:
1. Establishes the order of shifting and scaling in relationships involving both shifting and scaling operations.
2. Time-shifting takes precedence over time-scaling. In the above representation, the time shift b is performed first on x(t), resulting in an intermediate signal v(t) defined by
v(t) = x(t + b)
followed by time-scaling by a factor of a, that is,
y(t) = v(at) = x(at + b).
3. Example: Sketch y(t) = x(2t + 3) for x(t) given in (a)
)()( batxty +=
![Page 10: Monitoring Systems at Stationary Sources - US Environmental](https://reader036.fdocuments.us/reader036/viewer/2022071601/613d47ab736caf36b75b7469/html5/thumbnails/10.jpg)
10
Linear Transformations (4)
For the DT signal x[n] illustrated below:
Activity 4: Draw the following:(a) x[−n]; (b) x[2n]; (c) x[n + 3 ]; (d) x[2n + 3]
![Page 11: Monitoring Systems at Stationary Sources - US Environmental](https://reader036.fdocuments.us/reader036/viewer/2022071601/613d47ab736caf36b75b7469/html5/thumbnails/11.jpg)
11
Periodic Signals
1. Periodic signals: A periodic signal x(t) is a function of time that satisfies the condition x(t) = x(t + T) for all t
where T is a positive constant number and is referred to as the fundamental period of the signal.
Fundamental frequency (f) is the inverse of the period of the signal. It is measured in Hertz (Hz =1/s).2. Nonperiodic (Aperiodic) signals: are those that do not repeat themselves.
Activity 5: For the sinusoidal signals (a) x[n] = sin (5πn) (b) y[n] = sin(πn/3), determine the fundamental period N of the DT signals.
![Page 12: Monitoring Systems at Stationary Sources - US Environmental](https://reader036.fdocuments.us/reader036/viewer/2022071601/613d47ab736caf36b75b7469/html5/thumbnails/12.jpg)
12
Even vs. Odd Signals (1)
1. Even Signal: A CT signal x(t) is said to be an even signal if it satisfies the conditionx(−t) = x(t) for all t.
2. Odd Signal: The CT signal x(t) is said to be an odd signal if it satisfies the condition x(−t) = −x(t) for all t.
3. Even signals are symmetric about the vertical axis or time origin.4. Odd signals are antisymmetric (or asymmetric) about the time origin.
![Page 13: Monitoring Systems at Stationary Sources - US Environmental](https://reader036.fdocuments.us/reader036/viewer/2022071601/613d47ab736caf36b75b7469/html5/thumbnails/13.jpg)
13
Even vs. Odd Signals (2)
5. Signals that satisfy neither the even property nor the odd property can be divided into even and odd components based on the following equations:
Even component of x(t) = 1/2 [ x(t) + x(−t) ]Odd component of x(t) = 1/2 [ x(t) − x(−t) ]
Activity 6: For the signal
do the following:(a) sketch the signal(b) evaluate the odd part of the signal(c) evaluate the even part of the signal.
⎩⎨⎧ ≤≤−−= elsewhere0
20for |1|1)( tttx
![Page 14: Monitoring Systems at Stationary Sources - US Environmental](https://reader036.fdocuments.us/reader036/viewer/2022071601/613d47ab736caf36b75b7469/html5/thumbnails/14.jpg)
14
CT Exponential Signals (1)
CT exponential signals are of the form
where C and a can both be complex numbers.
atCetx =)(
.
complexcomplexGeneral Complex Exponential
imaginaryrealPeriodic Complex Exponential
realrealReal Exponential Signals
aCCharacteristic
![Page 15: Monitoring Systems at Stationary Sources - US Environmental](https://reader036.fdocuments.us/reader036/viewer/2022071601/613d47ab736caf36b75b7469/html5/thumbnails/15.jpg)
15
CT Exponential Signals: Real (2).
Real CT exponential signals:
where C and a are both positive numbers.
atCetx =)(
![Page 16: Monitoring Systems at Stationary Sources - US Environmental](https://reader036.fdocuments.us/reader036/viewer/2022071601/613d47ab736caf36b75b7469/html5/thumbnails/16.jpg)
16
CT Exponential Signals: Periodic Complex (3).
1. Periodic complex exponential signals:
where C is a positive number but (a = j ωo) is an imaginary number.
2. Difficult to draw. Magnitude and phase are drawn separately.Activity 7: Show that the periodic complex signal x(t) = C exp(jωot) has a fundamental period given by:
atCetx =)(
οωπ= 2T
![Page 17: Monitoring Systems at Stationary Sources - US Environmental](https://reader036.fdocuments.us/reader036/viewer/2022071601/613d47ab736caf36b75b7469/html5/thumbnails/17.jpg)
17
CT Exponential Signals: Sinusoidal (4).
1. Sinusoidal Signals:
where C is real and a is complex.2. Types of Sinusoidal Signals:
}{IM)(or}{RE)( atat CetxCetx ==
⎩⎨⎧
φ+ωφ+ω=
ο
ο waveSine)sin(
waveCosine)cos()( tCtCtx
![Page 18: Monitoring Systems at Stationary Sources - US Environmental](https://reader036.fdocuments.us/reader036/viewer/2022071601/613d47ab736caf36b75b7469/html5/thumbnails/18.jpg)
18
CT Exponential Signals: Sinusoidal (5).
Activity 8: Sketch the following sinusoidal signal
What is the value of the magnitude, fundamental frequency, fundamental phase, and power for the sinusoidal signal.
Activity 9: Show that the power of a sinusoidal signal
is given by A2 / 2.
)2/10sin(6)( π+π= ttx
)2sin()( tfAtx οπ=
![Page 19: Monitoring Systems at Stationary Sources - US Environmental](https://reader036.fdocuments.us/reader036/viewer/2022071601/613d47ab736caf36b75b7469/html5/thumbnails/19.jpg)
19
CT Exponential Signals: Complex (6).
1. General Complex Exponential signals:
where C and a are both complex numbers.2. By substituting C = |C|ejθ and a = (r + jωo), the magnitude and phase of x(t) can be expressed as
atCetx =)(
)()(:Phase|||)(|:Magnitude
φ+ω==
οttxeCtx rt
p
![Page 20: Monitoring Systems at Stationary Sources - US Environmental](https://reader036.fdocuments.us/reader036/viewer/2022071601/613d47ab736caf36b75b7469/html5/thumbnails/20.jpg)
20
CT Exponential Signals (7).
Activity 10: Derive and plot the magnitude and phase of the composite signal
tjtj eetx 35.2)( +=
![Page 21: Monitoring Systems at Stationary Sources - US Environmental](https://reader036.fdocuments.us/reader036/viewer/2022071601/613d47ab736caf36b75b7469/html5/thumbnails/21.jpg)
21
DT Exponential Signals (1).
nCnx α=][
DT exponential signals are of the form
where C and α can both be complex numbers.
Complex ComplexGeneral Complex Exponential
Imaginary (α = jω)RealSinusoidal Signals
RealRealReal Exponential Signals
αCCharacteristic
![Page 22: Monitoring Systems at Stationary Sources - US Environmental](https://reader036.fdocuments.us/reader036/viewer/2022071601/613d47ab736caf36b75b7469/html5/thumbnails/22.jpg)
22
DT Exponential Signals: Real (2).
nCnx β=][
1. For real exponential signals, C and α are both real
2. Depending upon the value of a, a different waveform is produced.α > 1 gives a rising exponentialα = 1 gives a constant line0 < α < 1 gives a decaying exponential−1 < α < 0 gives an alternating sign decaying exponentialα < −1 gives an alternating sign rising exponential
![Page 23: Monitoring Systems at Stationary Sources - US Environmental](https://reader036.fdocuments.us/reader036/viewer/2022071601/613d47ab736caf36b75b7469/html5/thumbnails/23.jpg)
23
DT Exponential Signals: Sinusoidal (3).
1. Sinusoidal Signals:
where C is real and α is complex.2. Types of Sinusoidal Signals:
3. Not all DT sinusoidal signals are periodic
}{IM)(or}{RE][ nn CtxCnx α=α=
⎩⎨⎧
φ+ωφ+ω=
ο
ο waveSine)sin(
waveCosine)cos(][ nCnCnx
![Page 24: Monitoring Systems at Stationary Sources - US Environmental](https://reader036.fdocuments.us/reader036/viewer/2022071601/613d47ab736caf36b75b7469/html5/thumbnails/24.jpg)
24
DT Exponential Signals: Sinusoidal (4).
4. Condition for periodicity: A DT sinusoidal signal
is periodic if (ωo / 2π) is a rational number with the period given by
Activity 11: Consider the signal
x[n] =cos(πn/3) sin(πn/6).
Determine if the signal is periodic. If yes, calculate the period N.
⎩⎨⎧
φ+ωφ+ω=
ο
ο waveSine)sin(
waveCosine)cos(][ nCnCnx
οωπ= mN 2
![Page 25: Monitoring Systems at Stationary Sources - US Environmental](https://reader036.fdocuments.us/reader036/viewer/2022071601/613d47ab736caf36b75b7469/html5/thumbnails/25.jpg)
25
Unit Step Function
DT domain:
CT domain
Activity: For the discrete time signal
Describe x[n] as a function of two step functions. Ans: U[n + 5] − U[n − 10]
⎩⎨⎧
<≥= 00
01][ nnnU
⎩⎨⎧
<≥= 00
01)( tttU
⎩⎨⎧ ≤≤−= elsewhere0
105for 1][ nnx
![Page 26: Monitoring Systems at Stationary Sources - US Environmental](https://reader036.fdocuments.us/reader036/viewer/2022071601/613d47ab736caf36b75b7469/html5/thumbnails/26.jpg)
26
Unit Sample (Impulse) Function
DT domain:
CT domain: Impulse function is defined as
⎩⎨⎧
≠==δ 00
01][ nnn
∫
∫
∞∞− οο
οοο
∞∞−
=−δ
−δ=−δ
δ=δ
=δ
)()()(.4)()()()(.3
)(||1)(.2
1)(.1
txdttttxtttxtttx
taatdtt
![Page 27: Monitoring Systems at Stationary Sources - US Environmental](https://reader036.fdocuments.us/reader036/viewer/2022071601/613d47ab736caf36b75b7469/html5/thumbnails/27.jpg)
27
Gate Function
DT domain:
CT domain
⎩⎨⎧ ≥=Π
elsewhere02/1][ Nnn
⎩⎨⎧ <=Π
elsewhere01)( Ttt
![Page 28: Monitoring Systems at Stationary Sources - US Environmental](https://reader036.fdocuments.us/reader036/viewer/2022071601/613d47ab736caf36b75b7469/html5/thumbnails/28.jpg)
28
Ramp Function
DT domain:
CT domain
⎩⎨⎧ ≥= elsewhere0
0][ nnnr
⎩⎨⎧ ≥= elsewhere0
0)( tttr