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 5-38. Optional: see references at end of lecture. Lab Fourier Analysis Do prelab before lab starts. PHSX 6 Spring Lecture 5 Page
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
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
Sine Wave in Time Domain Vt () Asin ft.5 Period f frequency period A V PP V pp Amplitude.5 -.5 - Vrms V() t -.5 3 4 5 6 milliseconds P Vt () Vrms A Power R R R For sine wave only PHSX 6 Spring Lecture 5 Page 4
Sine Wave in Frequency Domain Frequency period Frequency Domain "Stick Plot" frequency. V=A Amplitude.8.6.4. 3 4 5 6 Hertz Amplitude-Spectrum Plot PHSX 6 Spring Lecture 5 Page 5
Fourier Domain and Filtering Frequency Domain "Stick Plot". Amplitude.8.6.4. 3 4 5 6 Hertz Amplitude-Spectrum Plot Overlaid by Gain-Frequency PHSX 6 Spring Lecture 5 Page 6
Filtered Signal Frequency Domain "Stick Plot". Amplitude.8.6.4. 3 4 5 6 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
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
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
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
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
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
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
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
Fourier b T / b V()sin( t nt T ) dt n T T / n= b PHSX 6 Spring Lecture 5 Page 5
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
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
Constructing a Square Wave 4A Vt ( ) sin( t) sin(3 t) sin(5 t) 3 5.5.5.5.5.5.5 Amplitude -.5 Amplitude -.5 Amplitude -.5 - - - Amplitude -.5..8.6.4 Frequency Domain "Stick Frequency Plot" Domain "Stick Frequency Plot" Domain "Stick Plot" -.5 -.5.. 3 4 35 46 3 5 4 6 5 6 Amplitude.8.6.4 milliseconds Amplitude.8.6.4 milliseconds milliseconds... 5 5 5 5 5 5 3 5 5 3 5 3 Hertz Hertz Hertz PHSX 6 Spring Lecture 5 Page 8
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
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
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
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
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
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
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
Helpful applet: Fourier in the Audio http://www.falstad.com/fourier/ PHSX 6 Spring Lecture 5 Page 6
References. Paul Horowitz and Winfield Hill (989). The Art of Electronics, nd Ed., Cambridge, pages 5-38.. 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, http://www.falstad.com/mathphysics.html, 5 Feb 4. Efunda, Engineering Fundamentals web site; accessed 5 Feb http://www.efunda.com/designstandards/sensors/methods/dsp_nyquist.cfm 5. The Fundamentals of FFT-Based Signal Analysis and Measurment in LabVIEW and LabWindows ; 5 Feb, http://zone.ni.com/devzone/cda/tut/p/id/478 PHSX 6 Spring Lecture 5 Page 7