The (Fast) Fourier Transform

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2 The (Fast) Fourier Transform The Fourier transform (FT) is the analog, for non-periodic functions, of the Fourier series for periodic functions can be considered as a Fourier series in the limit that the period becomes infinite The Fast Fourier Transform (FFT) is a computer algorithm to calculate a FT for a discrete (or digitized) function input is a series of 2 p (complex) numbers representing a time function; output is 2 p (complex) numbers representing the coefficients at each frequency has a few rules to be obeyed Excel (or Maple/Mathmatica) will do this for you - it s not too hard to learn.

3 t 1 Δt 2 Δt 3 Δt 4 Δt 5 Δt 6 Δt 7 Δt 8 Δt 9 Δt 10 Δt 11 Δt 12 Δt 13 Δt 1024 Δt F(t) Time Δt Τ = ΝΔt

4 1 T = f 0 ; 2! T = " 0 Fundamental frequency (small!) Δt Δt 1 2!t = f N; 2" 2!t = # N 2Δt is SMALLEST period of a sinusoidal function that is sensible to consider - faster oscillations have no meaning for this function Only HALF the frequency spectrum is unique information Nyquist frequency

5 Aliasing samples function y x 4

6 The FFT c(ω) Δω = ω ω ω Ν

7 V t I = YV V 0 cos 0 t Y Vt V 0 cos2 0 t I V t V 0 cos 3 0 t t

8 Response of a damped oscillator to a periodic driving force of arbitrary shape: If a system is a LINEAR system, it means that if several sinusoidal driving forces are added and applied at once, then the response is just the sum of the individual responses. It seems obvious that the circuit is linear, but there are many systems that are non-linear. The following page shows this graphically for the same circuit the driving voltage that is the sum of the three previous ones, and the resulting current which is also the sum of the three previous currents. No longer pure sinusoids! Notice that the shape of the current and the voltage are not the same anymore! It s not true that there s one simple scaling factor and one phase shift!

9 V t V 0 cos 0 t V 0 cos2 0 t V 0 cos3 0 t Y V app I t

10 V t V 0 cos 0 t V 0 cos2 0 t V 0 cos3 0 t Y I Y V app I t It V 0 Y 0 cos 0 t 0 V 0 Y 20 cos2 0 t 20 V 0 Y 30 cos3 0 t 30

11 Forced, damped oscillator: the pulsed LRC circuit L I V ext C R V=IR This response function contains information about the circuit response at all frequencies. In fact the Fourier transform of this function is the admittance. We can use this to determine the admittance experimentally in a single measurement!!

12 A delta function in time is a superposition of equal mixes of sinusoids of all frequencies --> --> 30 terms --->

13 FT - you know this time frequency Black box Z(ω) Observe what (LRC) black box does to an impulse function Black box Z(ω) This was harmonic response expt - you know what black box (LRC) does to a single freq? Are these connected by FT?? They d better be - you find out!

14 Forced, damped oscillator: the pulsed LRC circuit I V ext L C R AFTER the pulse, we know that the charge decays (t=0 is the moment the pulse ends) V q(t) Qe t cos 1 t V 0 The pulse establishes the initial charge and current for the subsequent oscillating decay [i.e. q(0) and I(0)]. t R 2L 0 1 LC

15 Forced, damped oscillator: the pulsed LRC circuit I V ext V 0 L C R V BEFORE the pulse (or at the instant of the start of the pulse), there is no charge (no time to build up) no voltage across C there is no current (inductor kills it) no voltage across R ALL voltage is across inductor! di V 0 L dt t

16 Forced, damped oscillator: the pulsed LRC circuit I V ext V 0 L C R V I di' V 0 L 0 di' V 0 L dt t s t t s dt I(0) V 0 L t t s t=0 t s t q(0) V 0 L t 2 2

17 Forced, damped oscillator: the pulsed LRC circuit L q(t) Qe t cos t 1 I V ext C R V I(0) V 0 L t V 0 q(0) 0 q(t) V 0 1 L tet sin 1 t t s t=0 t s t I(t) V 0 L tet cos 1 t sin 1 t 1

18 Forced, damped oscillator: the pulsed LRC circuit I V ext L C R V I(t) V 0 time -> t s t=0 t s t I(t) V 0 L tet cos 1 t sin 1 t 1