ecen3513 signal analysis lecture #9 11 september 2006 n read section 2.7, 3.1, 3.2 (to top of page...

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


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


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


0

1.5

-1.50 1.0

0

1.5

-1.50 1.0

1 vp5 Hz

1/3 vp15 Hz


0

1.5

-1.50 1.0

1/5 vp25 Hz

0

1.5

-1.50 1.0

5 Hz+

15 Hz


0

1.5

-1.50 1.0

1/7 vp35 Hz

0

1.5

-1.50 1.0

5 Hz+

15 Hz+

25 Hz


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


5 cycle per second square wave generated using 50 sinusoids.

0

1.5

-1.50 1.0


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


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


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)


f (Hz)

∑e-jn2πf/T

n = 0

n = 0

0 5 10 15

1



f (Hz)

∑e-jn2πf/T

n = -1

n = +1

0 5 10 15

3



f (Hz)

∑e-jn2πf/T

n = -5

n = +5

0 5 10 15

11



f (Hz)

∑e-jn2πf/T

n = -10

n = +10

0 5 10 15

21



f (Hz)

∑e-jn2πf/T

n = -100

n = +100

0 5 10 15

201



f (Hz)

∑e-jn2πf/T

n = -100

n = +100

2.5 5

20

ecen3513 signal analysis lecture #9 11 september 2006 n read section 2.7, 3.1, 3.2 (to top of page...

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