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ECEN3513 Signal AnalysisECEN3513 Signal AnalysisLecture #9 11 September 2006Lecture #9 11 September 2006ECEN3513 Signal AnalysisECEN3513 Signal AnalysisLecture #9 11 September 2006Lecture #9 11 September 2006
Read section 2.7, 3.1, 3.2 (to top of page 6)Read section 2.7, 3.1, 3.2 (to top of page 6) Problems: 2.7-3, 2.7-5, 3.1-1Problems: 2.7-3, 2.7-5, 3.1-1
ECEN3513 Signal AnalysisECEN3513 Signal AnalysisLecture #10 13 September 2006Lecture #10 13 September 2006ECEN3513 Signal AnalysisECEN3513 Signal AnalysisLecture #10 13 September 2006Lecture #10 13 September 2006
Read section 3.4Read section 3.4 Problems: 3.1-5, 3.2-1, 3.3a & cProblems: 3.1-5, 3.2-1, 3.3a & c Quiz 2 results: Hi = 10, Low = 5.5, Ave = 7.88Quiz 2 results: Hi = 10, Low = 5.5, Ave = 7.88
Standard Deviation = 1.74Standard Deviation = 1.74
MathCad Correlation SolutionMathCad Correlation Solutionx t( ) 5 t t( ) 7 t( ) rxy1 valid when -7 < < 0.
y t( ) 4 t t( ) 10 t( )rxy1
7
t20 t2 t
d
Rxy 8
11
tx t( ) y t
d Rxy 6.7( ) 6.21rxy1 6.7( ) 6.21
i 0 1000 i 20i
25 t
i20
i
25
20 10 0 10 200
2000
4000
Rxy i
i
MathCad Correlation SolutionMathCad Correlation Solutionx t( ) 5 t t( ) 7 t( ) rxy1 valid when -7 < < 0.
y t( ) 4 t t( ) 10 t( )rxy1
7
t20 t2 t
d
Rxy
tx t( ) y t
d Rxy 6.7( ) 0rxy1 6.7( ) 6.21
i 0 1000 i 20i
25 t
i20
i
25
20 10 0 10 200
2000
4000
Rxy i
i
MathCad Correlation SolutionMathCad Correlation Solutionx t( ) 5 t t( ) 7 t( ) rxy1 valid when -7 < < 0.
y t( ) 4 t t( ) 10 t( )rxy1
7
t20 t2 t
d
Rxy 20
20
tx t( ) y t
d Rxy 6.7( ) 0rxy1 6.7( ) 6.21
i 0 1000 i 20i
25 t
i20
i
25
20 10 0 10 200
2000
4000
Rxy i
i
MathCad Convolution SolutionMathCad Convolution SolutionActual equation for z(t).Valid when 10 < t< 17.x t( ) 5 t t( ) 7 t( )
y t( ) 4 t t( ) 10 t( )
zactual t( )t 7
10
20 t 2
dz t( )
.01
17.01
x t y
d z 16.7( ) 404.865
zactual 16.7( ) 404.79
i 0 1000 i 5i
25 t
i20
i
25
20 10 0 10 200
2000
4000
z ti
ti
MathCad Correlation SolutionMathCad Correlation SolutionActual equation for z(t).Valid when 10 < t< 17.x t( ) 5 t t( ) 7 t( )
y t( ) 4 t t( ) 10 t( )
zactual t( )t 7
10
20 t 2
dz t( )
1
18
x t y
d z 16.7( ) 0
zactual 16.7( ) 404.79
i 0 1000 i 5i
25 t
i20
i
25
20 10 0 10 200
2000
4000
z ti
ti
10 0 10 20 30 400
20
40
y i
i
0
1000
j
y j 25
200.8
Time 16.7
10 0 10 20 30 400
20
40
x Time i
i
0
1000
j
x j 25
123.2
10 0 10 20 30 400
1000
2000
y i x Time i
i
0
1000
j
y j x Time j
25
431.578
z 16.7( ) 0
You can't always trust your software
tools!
You can't always trust your software
tools!
Generating a Square Wave...Generating a Square Wave...
5 cycle per second square wave.
0
1.5
-1.50 1.0
Generating a Square Wave...Generating a Square Wave...
0
1.5
-1.50 1.0
0
1.5
-1.50 1.0
1 vp5 Hz
1/3 vp15 Hz
Generating a Square Wave...Generating a Square Wave...
0
1.5
-1.50 1.0
1/5 vp25 Hz
0
1.5
-1.50 1.0
5 Hz+
15 Hz
Generating a Square Wave...Generating a Square Wave...
0
1.5
-1.50 1.0
1/7 vp35 Hz
0
1.5
-1.50 1.0
5 Hz+
15 Hz+
25 Hz
Generating a Square Wave...Generating a Square Wave...
0
1.5
-1.50 1.0
5 Hz+
15 Hz+
25 Hz+
35 Hz
cos2*pi*5t - (1/3)cos2*pi*15t + (1/5)cos2*pi*25t - (1/7)cos2*pi*35t)
5 cycle per second square wave generated using 4 sinusoids
Generating a Square Wave...Generating a Square Wave...
5 cycle per second square wave generated using 50 sinusoids.
0
1.5
-1.50 1.0
Generating a Square Wave...Generating a Square Wave...
5 cycle per second square wave generated using 100 sinusoids.
0
1.5
-1.50 1.0
Fourier Series(Trigonometric Form)Fourier Series(Trigonometric Form)
x(t) = ax(t) = a00 + + ∑ (a∑ (anncos ncos nωω00t + bt + bnnsin nsin nωω00t)t)
aa00 = (1/T) x(t)1 dt = (1/T) x(t)1 dt
aann = (2/T) x(t)cos n = (2/T) x(t)cos nωω00t dtt dt
bbnn = (2/T) x(t)sin n = (2/T) x(t)sin nωω00t dtt dt
n=1
∞
T
T
T
x(t) periodicT = periodω0 = 2π/T
Fourier Series(Harmonic Form)Fourier Series(Harmonic Form)
x(t) = ax(t) = a00 + + ∑ c∑ cnncos(ncos(nωω00t - t - θθnn))
aa00 = (1/T) x(t)1 dt = (1/T) x(t)1 dt
ccnn22 = a = ann
22 + b + bnn22
θθnn = tan = tan-1-1(b(bnn/a/ann))
n=1
∞
T
x(t) periodicT = periodω0 = 2π/T
Fourier Series(Exponential Form)Fourier Series(Exponential Form)
x(t) = x(t) = ∑ d∑ dnnee jn jnωω00tt
ddnn = (1/T) x(t)e = (1/T) x(t)e-jn-jnωω00tt dt dt
n= -∞
∞
T
x(t) periodicT = periodω0 = 2π/T
TransformsTransforms
X(s) = x(t) e-st dt
0-
∞ Laplace
∞
X(3) = x(t) e-3t dt
0-
TransformsTransforms
X(f) = x(t) e-j2πft dt
-∞
∞ Fourier
ee-j2-j2ππftft = cos( = cos(22ππft) - j sin(ft) - j sin(22ππft)ft) Re[X(f)] = similarity between cos(Re[X(f)] = similarity between cos(22ππft) & x(t)ft) & x(t)
Re[X(10.32)] = amount of 10.32 Hz cosine in x(t)Re[X(10.32)] = amount of 10.32 Hz cosine in x(t) Im[X(f)] = similarity between sin(Im[X(f)] = similarity between sin(22ππft) & x(t)ft) & x(t)
-Im[X(10.32)] = amount of 10.32 Hz sine in x(t)-Im[X(10.32)] = amount of 10.32 Hz sine in x(t)
Fourier SeriesFourier Series
Time Domain signal Time Domain signal mustmust be periodic be periodic Line SpectraLine Spectra
Energy only at discrete frequenciesEnergy only at discrete frequenciesFundamental (1/T Hz)Fundamental (1/T Hz)Harmonics (n+1)/T Hz; n = 1, 2, 3, ...Harmonics (n+1)/T Hz; n = 1, 2, 3, ...
Spectral envelope is based on FT of Spectral envelope is based on FT of periodic base functionperiodic base function
Fourier TransformsFourier Transforms
X(f) = x(t) e-j2πft dt
-∞
∞ Forward
x(t) = X(f) ej2πft df
-∞
∞ Inverse
Fourier TransformsFourier Transforms
X(ω) = x(t) e-jωt dt
-∞
∞ Forward
x(t) = X(ω) ejωt dω 2π
-∞
∞ Inverse
Fourier TransformsFourier Transforms Basic TheoryBasic Theory How to evaluate simple integral transformsHow to evaluate simple integral transforms How to use tablesHow to use tables
On the job (BS signal processing or Commo)On the job (BS signal processing or Commo) Mostly you’ll use Fast Fourier TransformMostly you’ll use Fast Fourier Transform
Info above will help you spot errorsInfo above will help you spot errors Only occasionally will you find FT by handOnly occasionally will you find FT by hand
Masters or PhD may do so more oftenMasters or PhD may do so more often
Summing Complex Exponentials(T = 0.2 seconds)
Summing Complex Exponentials(T = 0.2 seconds)
f (Hz)
∑e-jn2πf/T
n = 0
n = 0
0 5 10 15
1
Summing Complex Exponentials(T = 0.2 seconds)
Summing Complex Exponentials(T = 0.2 seconds)
f (Hz)
∑e-jn2πf/T
n = -1
n = +1
0 5 10 15
3
Summing Complex Exponentials(T = 0.2 seconds)
Summing Complex Exponentials(T = 0.2 seconds)
f (Hz)
∑e-jn2πf/T
n = -5
n = +5
0 5 10 15
11
Summing Complex Exponentials(T = 0.2 seconds)
Summing Complex Exponentials(T = 0.2 seconds)
f (Hz)
∑e-jn2πf/T
n = -10
n = +10
0 5 10 15
21
Summing Complex Exponentials(T = 0.2 seconds)
Summing Complex Exponentials(T = 0.2 seconds)
f (Hz)
∑e-jn2πf/T
n = -100
n = +100
0 5 10 15
201
Summing Complex Exponentials(T = 0.2 seconds)
Summing Complex Exponentials(T = 0.2 seconds)
f (Hz)
∑e-jn2πf/T
n = -100
n = +100
2.5 5
20