Phsx262 S11 Lec05 - Solar physicssolar.physics.montana.edu/.../RBnotes/phsx262_S11_Lec05.pdf · 2012-01-06 · Based with permission on lectures by John Getty PHSX 262 Spring 2011

Based with permission on lectures by John GettyPHSX 262 Spring 2011 Lecture 5 Page 1

Today2/15/11 Lecture 5Fourier Series

• Time-Frequency Decomposition/Superposition• Fourier Components (Ex. Square wave)• Filtering• Spectrum Analysis

– Windowing– Fast Fourier Transform– Sweep Frequency Analyzer

Homework: (due next Tuesday)1) Write down the expected powers and dBVs for the 3rd harmonic of all four functions in the lab if

they were 2 Vpp functions (versus 1Vpp functions).2) For a square wave of period 300 microseconds that goes from -1 volts to +1 volts into 50 ohms,

what are the frequencies and powers in the 4 strongest frequency components? Does it matter how square wave is centered in time (i.e. odd or even with respect to t=0)?

3) How much power in watts is dissipated into a 50 ohm resistor by a -13dBV signal?4) What is the ratio of the powers and the voltages of a -27dBV signal and a -33dBV signal?

Reading• See Prelab• Horowitz and Hill 2nd Ed., pages 1025-1038.• Optional: see references at end of lecture.

LabFourier AnalysisDo prelab before lab starts.


Fourier’s TheoremFrench mathematician Joseph Fourier

(1768-1830), discovered that he could represent any real functions with a series of weighted sines and cosines.

In circuit analysis we use Fourier’s Theorem to “decompose” a complex time domain signal into its discrete sinusoidal parts (the frequency domain.) Superposition of these frequency component returns the signal to the time domain.


The Time and Frequency DomainsAmplitude

(not power)

Time domainMeasurements(Oscilloscope)

Frequency DomainMeasurements

(Spectrum Analyzer)

Phase(or delay)


Sine Wave in Time Domain

-1.5

-1

-0.5

0

0.5

1

1.5

0 100 200 300 400 500 600milliseconds

Am

plitu

de

Vpp

Period

2PPVA

1f frequency

period( ) sin2V t A ft

2 2 2( )

2rmsV t V AP Power

R R R

2( )rmsV V t

For sine wave only


Frequency Domain "Stick Plot"

0

0.2

0.4

0.6

0.8

1

1.2

0 10 20 30 40 50 60

Hertz

Am

plitu

de

Sine Wave in Frequency Domain

V=A

Frequency1period

frequency

Amplitude-Spectrum Plot



0

0.2

0.4

0.6

0.8

1

1.2

0 10 20 30 40 50 60

Hertz

Am

plitu

de

Fourier Domain and Filtering

Amplitude-Spectrum Plot Overlaid by Gain-Frequency


Filtered Signal

Each component is transmitted at its filtered amplitude.Filter can also introduce phase shift of each component.Resultant signal is the sum of the transmitted components.


0

0.2

0.4

0.6

0.8

1

1.2

0 10 20 30 40 50 60Hertz

Am

plitu

de


With permission, Agilent Technologies

0 0 01

( ) ( cos sin )n nn

V t a a n t b n t

ACDC

Fourier Series

2 2

2n n

n

a bP

R

(for periodic functions)

20 0 /P a R

Power in harmonics


dBVSince scope only measures voltage and doesn’t know what load resistor you are using, it can’t measure power absolutely, so if measure in dBV

dBV is a measure of relative POWER (not voltage)!!!!A 20 dBV sinewave has 100 times more power than 0dBV

and 10 times the voltage.

dBV is relative to the power of a sinewave relative to a 1 Volt RMS sinewave signal.

dBV = 10 log10(<V2> / 1Vrms) = 10 log10(A2/2) where A is the amplitude of the sinewave in volts

Note dBV is it independent of resistive load.


Fourier Transform (Decomposition)0 0 0

1

( ) ( cos sin )n nn

V t a a n t b n t

Fourier series:

0

0

2

00 2

1 ( )T

T

a V t dtT

0

0

2

00 2

2 ( )cos(2 )T

nT

a V t nt T dtT

0

0

2

00 2

2 ( )sin(2 )T

nT

b V t nt T dtT

DC

Even part of

V(t)

Odd part of

V(t)

00

2T period


Odd and Even Symmetrycos(x)even sym.

( ) ( )f x f x

sin(x)odd sym.

( ) ( )f x f x


Fourier’s a0 for a Square WaveT0

0

0

0 / 2( )

- / 2 0A t T

V tA T t

0

0

/ 2

00 / 2

1 ( )T

T

a V t dtT

0

0

2 0

0 00 2

1 1 ( )T

T

Adt A dtT T

0 0

0

02 2

A T TT

A

A

For this waveform, DC component is precisely zero


Fourier’s an for a Square Wave0

0

/ 2

00 / 2

2 ( )cos(2 )T

nT

a V t nt T dtT

( ) is odd 0 for all nnV t a


n=1

14Ab

Fourier’s bn for a Square Wave0

0

/ 2

00 / 2

2 ( )sin(2 )T

nT

b V t nt T dtT


n=2

Fourier b2

2 0b

0

0

/ 2

00 / 2

2 ( )sin(2 )T

nT

b V t nt T dtT


n=3

Fourier b3

13 3

bb

0

0

/ 2

00 / 2

2 ( )sin(2 )T

nT

b V t nt T dtT

343

Ab


Fourier Series for Square Wave

0 0 04 1 1( ) sin( ) sin(3 ) sin(5 )

3 5AV t t t t

0 0 01

( ) ( cos sin )n nn

V t a a n t b n t

Fourier’s infinite series:

For a square wave centered around ground and time=0:

Fundamental Third HarmonicFifth Harmonic

0 0; 0 (odd function); 4 (n odd) 0 (n even)

n

n n

a aAb b

n


-1.5

-1

-0.5

0

0.5

1

1.5

0 100 200 300 400 500 600milliseconds

Am

plitu

de


0

0.2

0.4

0.6

0.8

1

1.2

0 5 10 15 20 25 30

Hertz

Am

plitu

de

Constructing a Square Wave

0 0 04 1 1( ) sin( ) sin(3 ) sin(5 )

3 5AV t t t t

-1.5

-1

-0.5

0

0.5

1

1.5

0 100 200 300 400 500 600milliseconds

Am

plitu

de


0

0.2

0.4

0.6

0.8

1

1.2

0 5 10 15 20 25 30

Hertz

Am

plitu

de

-1.5

-1

-0.5

0

0.5

1

1.5

0 100 200 300 400 500 600milliseconds

Am

plitu

de


0

0.2

0.4

0.6

0.8

1

1.2

0 5 10 15 20 25 30

Hertz

Am

plitu

de


Some Fourier Coefficients1

1Thomas and Rosa (2004). “The Analysis and Design of Linear Circuits,” 4th Ed., John Wiley and Sons, Inc


More Fourier Coefficients1

1Thomas and Rosa (2004). “The Analysis and Design of Linear Circuits,” 4th Ed., John Wiley and Sons, Inc


Computing Discrete Fourier Transforms

If there are N sampled points per period in time domain Requires N Fourier components to fully represent

Components an and bn count as one Fourier frequency component Components can be expressed as

A() =|A()|exp(i()) A() is complex

Requires N x N complex multiplies to compute disceteFourier series of N sample long time series.

Fast Fourier Transform (FFT)- Use math tricks to minimize number of multiplies N log2 (N) multiplies to compute Fourier Series Your scopes do FFTs


Filtered signal

0 0 01

( ) ( cos sin )n nn

V t a a n t b n t

Fourier series:

0 0 0 01

( ) (0) ( )( cos sin )out n nn

V t a T T nf a n t b n t

Assume V(t) is filtered by filter T(f) to produce Vout(t)

0

0 0 0 0 01

( ) (0)

( ) cos ( ) sin ( )

out

n nn

V t a A

A nf a n t nf b n t nf

If filter T(f) is real

If filter T(f) is complex: A(f)exp(j(f))

0 0 /(2 )f


Hanning Window

Windows for the FFTRectangular

Window(Boxcar)

Discontinuitiescreate sidebands

Smooth up and down limits sidebands


Center frequency of a hi-Q filter is swept across the frequency band.

Could miss components that come and go, like frequency hopper.

Swept Spectrum Analyzer

Good for high frequency signals. Typically expensive. Depends on signals being repetitive.

f0 3f0 5f0

Pow

er

freq


Behaves like simultaneous parallel filters: Does miss any non-constant components.

Pow

erFast Fourier Transform Analyzer

Captures full signal, but limited in bandwidth. Low cost. Built into some oscilloscopes.

f0 3f0 5f0 freq

Time domain signal is first digitized, then FFT is performed


Fourier in the Audio

http://www.falstad.com/fourier/

Helpful applet:


References1. Paul Horowitz and Winfield Hill (1989). “The Art of Electronics,” 2nd Ed., Cambridge,

pages 1025-1038.2. Roland E. Thomas and Albert J. Rosa (2004). “The Analysis and Design of Linear

Circuits,” 4TH Ed., John Wiley and Sons3. Paul Falstad, “some applets … to help visualize various concepts in math and

physics”, http://www.falstad.com/mathphysics.html, 15 Feb 20104. “Efunda, Engineering Fundamentals” web site; accessed 15 Feb 2010

http://www.efunda.com/designstandards/sensors/methods/DSP_nyquist.cfm5. “The Fundamentals of FFT-Based Signal Analysis and Measurment in LabVIEW

and LabWindows”; 15 Feb 2010, http://zone.ni.com/devzone/cda/tut/p/id/4278

Documents

Phsx262 S11 Lec05 - Solar physicssolar.physics.montana.edu/.../RBnotes/phsx262_S11_Lec05.pdf · 2012-01-06 · Based with permission on lectures by John Getty PHSX 262 Spring 2011