Lab Fourier Analysis Do prelab before lab starts. PHSX 262 Spring 2011 Lecture 5 Page 1. Based with permission on lectures by John Getty

Transcription

1 Today /5/ Lecture 5 Fourier Series Time-Frequency Decomposition/Superposition Fourier Components (Ex. Square wave) Filtering Spectrum Analysis Windowing Fast Fourier Transform Sweep Frequency Analyzer Homework: (due next Tuesday) ) Write down the expected powers and dbvs for the 3rd harmonic of all four functions in the lab if they were Vpp functions (versus Vpp functions). ) For a square wave of period 3 microseconds that goes from - volts to + volts into 5 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=)? 3) How much power in watts is dissipated into a 5 ohm resistor by a -3dBV signal? 4) What is the ratio of the powers and the voltages of a -7dBV signal and a -33dBV signal? Reading See Prelab Horowitz and Hill nd Ed., pages Optional: see references at end of lecture. Lab Fourier Analysis Do prelab before lab starts. PHSX 6 Spring Lecture 5 Page

2 Fourier s Theorem French mathematician Joseph Fourier (768-83), 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. PHSX 6 Spring Lecture 5 Page

3 The Time and Frequency Domains Amplitude (not power) Phase (or delay) Time domain Measurements (Oscilloscope) Frequency Domain Measurements (Spectrum Analyzer) PHSX 6 Spring Lecture 5 Page 3

4 Sine Wave in Time Domain Vt () Asin ft.5 Period f frequency period A V PP V pp Amplitude Vrms V() t milliseconds P Vt () Vrms A Power R R R For sine wave only PHSX 6 Spring Lecture 5 Page 4

5 Sine Wave in Frequency Domain Frequency period Frequency Domain "Stick Plot" frequency. V=A Amplitude Hertz Amplitude-Spectrum Plot PHSX 6 Spring Lecture 5 Page 5

6 Fourier Domain and Filtering Frequency Domain "Stick Plot". Amplitude Hertz Amplitude-Spectrum Plot Overlaid by Gain-Frequency PHSX 6 Spring Lecture 5 Page 6

7 Filtered Signal Frequency Domain "Stick Plot". Amplitude Hertz 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. PHSX 6 Spring Lecture 5 Page 7

8 Fourier Series (for periodic functions) () ( cos sin ) V t a a n t b n t n n n DC P AC n a b n n R Power in harmonics P a / R With permission, Agilent Technologies PHSX 6 Spring Lecture 5 Page 8

9 dbv Since 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 dbv sinewave has times more power than dbv and times the voltage. dbv is relative to the power of a sinewave relative to a Volt RMS sinewave signal. dbv = log (<V > / Vrms) = log (A /) where A is the amplitude of the sinewave in volts Note dbv is it independent of resistive load. PHSX 6 Spring Lecture 5 Page 9

10 Fourier Transform (Decomposition) Fourier series: a T V() t a ( ancosntbnsin nt) n T T period V() t dt T T an V()cos( t nt T ) dt T T T bn V( t)sin( nt T ) dt T T DC Even part of V(t) Odd part of V(t) PHSX 6 Spring Lecture 5 Page

11 Odd and Even Symmetry cos(x) even sym. f ( x) f( x) sin(x) odd sym. f ( x) f( x) PHSX 6 Spring Lecture 5 Page

12 Fourier s a for a Square Wave V() t A t T / A T t - / T A A a T T / T / V() t dt T Adt ( A) dt T T T A T T T For this waveform, DC component is precisely zero PHSX 6 Spring Lecture 5 Page

13 Fourier s a n for a Square Wave T / a V()cos( t nt T ) dt n T T / V( t) is odd a for all n n PHSX 6 Spring Lecture 5 Page 3

14 Fourier s b n for a Square Wave T / b V()sin( t nt T ) dt n T T / n= b 4A PHSX 6 Spring Lecture 5 Page 4

15 Fourier b T / b V()sin( t nt T ) dt n T T / n= b PHSX 6 Spring Lecture 5 Page 5

16 Fourier b 3 T / b V()sin( t nt T ) dt n T T / n=3 b 3 4A 3 b 3 PHSX 6 Spring Lecture 5 Page 6 b 3

17 Fourier Series for Square Wave Fourier s infinite series: For a square wave centered around ground and time=: V() t a ( a cosn tb sin n t) n n n a ; an (odd function); 4A bn (n odd) bn (n even) n 4A V( t) sin( t) sin(3 t) sin(5 t) 3 5 Fundamental Third Harmonic Fifth Harmonic PHSX 6 Spring Lecture 5 Page 7

18 Constructing a Square Wave 4A Vt ( ) sin( t) sin(3 t) sin(5 t) Amplitude -.5 Amplitude -.5 Amplitude Amplitude Frequency Domain "Stick Frequency Plot" Domain "Stick Frequency Plot" Domain "Stick Plot" Amplitude milliseconds Amplitude milliseconds milliseconds Hertz Hertz Hertz PHSX 6 Spring Lecture 5 Page 8

19 Some Fourier Coefficients Thomas and Rosa (4). The Analysis and Design of Linear Circuits, 4th Ed., John Wiley and Sons, Inc PHSX 6 Spring Lecture 5 Page 9

20 More Fourier Coefficients Thomas and Rosa (4). The Analysis and Design of Linear Circuits, 4th Ed., John Wiley and Sons, Inc PHSX 6 Spring Lecture 5 Page

21 Computing Discrete Fourier Transforms If there are N sampled points per period in time domain Requires N Fourier components to fully represent Components a n and b n 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 discete Fourier series of N sample long time series. Fast Fourier Transform (FFT) - Use math tricks to minimize number of multiplies N log (N) multiplies to compute Fourier Series Your scopes do FFTs PHSX 6 Spring Lecture 5 Page

22 Fourier series: Filtered signal Assume V(t) is filtered by filter T(f) to produce V out (t) If filter T(f) is real V() t a ( a cosn tb sin n t) n n n V () t a T() T( nf )( a cosn tb sin n t) out n n n If filter T(f) is complex: A(f)exp(j(f)) V () t a A() out n f /( ) Anf ( ) a cos n t( nf) bsin n t( nf) n n PHSX 6 Spring Lecture 5 Page

23 Windows for the FFT Rectangular Window (Boxcar) Discontinuities create sidebands Smooth up and down limits sidebands Hanning Window PHSX 6 Spring Lecture 5 Page 3

24 Power Swept Spectrum Analyzer Center frequency of a hi-q filter is swept across the frequency band. Could miss components that come and go, like frequency hopper. f 3f 5f freq Good for high frequency signals. Typically expensive. Depends on signals being repetitive. PHSX 6 Spring Lecture 5 Page 4

25 Fast Fourier Transform Analyzer Time domain signal is first digitized, then FFT is performed Power Behaves like simultaneous parallel filters: Does miss any non-constant components. f 3f 5f freq Captures full signal, but limited in bandwidth. Low cost. Built into some oscilloscopes. PHSX 6 Spring Lecture 5 Page 5

26 Helpful applet: Fourier in the Audio PHSX 6 Spring Lecture 5 Page 6

27 References. Paul Horowitz and Winfield Hill (989). The Art of Electronics, nd Ed., Cambridge, pages Roland E. Thomas and Albert J. Rosa (4). The Analysis and Design of Linear Circuits, 4 TH Ed., John Wiley and Sons 3. Paul Falstad, some applets to help visualize various concepts in math and physics, 5 Feb 4. Efunda, Engineering Fundamentals web site; accessed 5 Feb 5. The Fundamentals of FFT-Based Signal Analysis and Measurment in LabVIEW and LabWindows ; 5 Feb, PHSX 6 Spring Lecture 5 Page 7