Introduction to Sampled Signals and Fourier Transforms

Transcription

1 Introduction to Sampled Signals and Fourier Transforms Physics116C, 4/28/06 D. Pellett References: Essick, Advanced LabVIEW Labs Press et al., Numerical Recipes, Ch. 12 Brigham, The Fast Fourier Transform and its Applications Any original presentations copyright D. Pellett

2 Sampled Signals and Nyquist Frequency As we were saying at the end of Physics 116B: 2

3 Sampling Theorem 3

4 Aliasing of f > fc In general, will need to limit bandwidth with a low-pass filter before digitizing. For audio response to 17 khz with 44 khz sample frequency must use sharp cutoff filter to eliminate f > fc = 22 khz (although little power may remain in this part of the spectrum). 4

5 Fourier Transform To pursue this we need Fourier transforms This is the convention used in Essick It has a nice correspondence with complex phasors from 116A 5

6 Some FT Theorems (again, same convention as Essick) Convolution integral is important concept Also useful in statistics (as is Fourier transform). More on this later 6

7 Convolution Theorem Again, we We will look at an example soon 7

8 Power Spectral Density in Frequency Define 8

9 Some Useful Transform Pairs Spacing of δ fcns in t is T while spacing in f is 1/T 9

10 Time Domain h(t) = A T 0 < t < T 0 = A 2 t = T 0, T 0 = 0 t < T 0, t > T 0 h(t) = 2Af 0 sinc(2πf 0 t) h(t) = K More Transform Pairs Frequency Domain H(f) = 2AT 0 sinc(2πt 0 f) H(f) = A f 0 < f < f 0 = A 2 f = f 0, f 0 = 0 f < f 0, f > f 0 H(f) = Kδ(f) h(t) = Kδ(t) H(f) = K h(t) = n= δ(t nt ) H(f) = n= 1 T δ(t n T ) h(t) = A cos(2πf 0 t) H(f) = A 2 δ(f f 0) + A 2 δ(f + f 0) h(t) = A sin(2πf 0 t) H(f) = i A 2 δ(f f 0) + i A 2 δ(f + f 0) h(t) = exp( at 2 ) H(f) = π a exp( πf 2 /a) (will provide this table on midterm) 10

11 Convolution Example 11

12 Convolution Example (continued) 12

13 Convolution Example (concluded) 13

14 Fourier Transform of Triangle Fcn 14

15 One-Sided Spectral Density 15

16 Sampled Waveforms as Series of Delta Functions (Ideal Case) We can represent a sampled waveform mathematically (ideal case) as the product of the waveform h(t) with an infinite sequence of δ functions spaced by the sampling period T=1/fs where fs is the sampling frequency Assume H(f) is bandwidth limited: H(f)=0 for f > fc = fs/2 The resulting Fourier transform (F.T.) of the sampled waveform is the convolution of H(f) with the F.T. of the sequence of δ fcns: another sequence of δ fcns spaced by 1/T = 2fc The original H(f) will extend between -fc and fc about f=0 But it will be replicated about every δ fcn in f (spaced by 2fc) 16

17 Multiply Complete H(f) by Boxcar(f) to Get Desired Range in f Select only the desired frequencies as follows: multiply the complete H(f) of the sampled waveform by a boxcar function in f extending from -fc to fc about f=0 The resulting h(t) is the convolution of the F.T. of the boxcar in the time domain with the original sampled waveform. This is the sampling theorem proof: 17

18 Alaising if H(f) not BW Limited The result is a distorted waveform due to the aliasing 18

19 Discrete Fourier Transform 19

20 Fast Fourier Transform Fast Fourier transform (FFT) uses a clever numerical algorithm to reduce a calculation which would have grown like N 2 (where N = number of samples hi to transform) to one which grows like N log2 N. This is a great help in the calculation. Usually called Cooley-Tukey algorithm but was noted earlier by Danielson and Lanczos, for example. The FFT can be used to calculate power spectra in the frequency domain and estimate amplitudes of particular frequency components. See Essick, Ch. 9 for details (will return to this). 20

21 Windowing One must take into account the effects of using a finite-length series of samples in our FFT (the sampling theorem assumed the series extended from to ). Using only a relatively short length of the waveform amounts to multiplying the series hi by a boxcar function (the window ) in the time domain whose width is the sample length. The resulting Fourier transform (freq. spectrum) is thus the convolution of the original F.T. with the F.T. of a boxcar function. If the original F.T. had sharp peaks, as with h(t)=acos(2πf0t), the result of the convolution is an H(f) with sinc-like peaks: the spectrum gets spread out by the sinc in f (called leakage ) The amount of leakage can be reduced by using a smoother windowing function. Each sample hj is multiplied by its weighting factor wj A good choice is the Hanning window (effect shown on next slide): w j = 1 2 [ 1 cos ( 2πj N 1 See the section, Leakage and Windowing, in Essick, Ch. 9 for details. )] 21

22 LabVIEW Windowing Example Use of Hanning window reduces leakage sufficiently to reveal a nearby weak signal Window: Square ( none ) Hanning Example from LabVIEW: LabVIEW 6 Student Edition/examples/analysis/windexmpl.lib/ Window Comparison 22

23 Filtering in the Frequency Domain To do frequency-dependent filtering on the sampled signal in the time domain we can apply a filter function to the FFT spectrum in the freq. domain (FFT, multiply by filter fcn, transform back). Can do low-pass, bandpass, band stop, high pass filters of various orders, etc. Can t eliminate aliasing this way, however The filter will have an effect on the time response much as we saw with analog filters in 116A (sharp edges in f can cause damped ringing in t) Real-time digital filter techniques can also be developed (will discuss later) If real-time results not needed, FFT method probably more straightforward Other related topics: deconvolution, correlation function Reference: Press, et al., Numerical Recipes, Sec

24 Next Experimental resolution functions and convolutions Relation to statistics Complex Fourier series and FFT Complex Fourier amplitudes Ak Connection with Essick notation and Ak=Hk/N Connection with power spectra ENBW 24