Outline Analysis of Waveforms and Transforms How many Samples to Take Aliasing Negative Spectrum Frequency Resolution Synchronizing Sampling Non-repetitive Waveforms Picket Fencing A Sampled Data System
Data Acquisition Systems Signal DAQ System The Answer?
Fourier Series for Repetitive Waves N N + + + pn cos( nω t) q qn sin( n t) n= n= x( t) = p ω Amplitude.5.5 -.5 - -.5 H H3 SUM 2 3 4 5 6 Time (sample number) P s & Q s determine the resulting wave
Finding the Frequency Components How much sinewave at frequency h.f is in signal s(t) S( h) = Corrolation[ s( t), sin(2π. f. h)] The correlation is carried out as follows: S ( h) = s( t). sin(2π. f. h) dt Similarly (to account for phase angles) C ( h) = s( t). cos(2π. f. h) dt C( h) Using complex number operations j.2π. f. h. t + j S( h) = s( t). e dt
Fourier Transforms = = 2 ) ( ) ( N n N n f j e n s N f S π The component in waveform s(t) at frequency f is found by a Fourier Transform: For sampled data, the discrete Fourier transform (DFT) of repetitive waveform: The Fast Fourier Transform (FFT) is a fast calculation method for the DFT = dt e t s f S t f j π 2 ) ( ) (
How Many Samples Are Needed? Sampling Theorem: The Sampling Frequency must be At Least Twice the Bandwidth of the System! So for a 5 Hz signal will Hz be OK?
Sampling a 5 Hz Signal Signal at 5Hz ADC To DSP Noise at.4 khz ( 28*5) Sampling Frequency =.6 khz, 32 samples/cycle
5 Hz Signal with Noise 5Hz Sine with 4Hz "Noise" Amplitude f n 2 3 4 5 n Time Now Sample at.6 khz, (32 times 5 Hz).
Sampling Too Slowly 32 Samples Per Cycle of Noisey Sinewave Amplitude 2 3 4 5 Time
Aliasing of.4 khz Noise in.6khz Sampling System True Spectrum.4 khz Noise Aliasing Line H.6 khz / 2 H28 H H6 H28 Wrapped Spectrum H H4 H6 Wrapped Around
Sampling Too Slowly 32 Samples Per Cycle of Noisey Sinewave Amplitude 2 3 4 5 Time
Use of an Anti-Aliasing Aliasing Filter Signal at 5Hz Low Pass Filter, -8db at Fs/2 ADC To DSP Noise at any frequency Sampling Frequency = Fs
Negative Spectrum Amplitude.2.8.6.4.2 -.2 5 5 Sample Number 6 Point FFT a+jb a-jb Negative frequencies Use Hilbert Transform to Cancel ve Spectrum
Negative Frequencies Inverse FFT k = x ( t) = X exp( jkωt) k For k negative, we get Negative Frequencies As the negatives are conjugates of the positives, they cancel the imaginary parts and give a real series x(t)
Inverse FFT IFFT Time Series is Complex!
Inverse FFT Time Series is Real IFFT Complete the ve half of the spectrum (Use complex conjugate).5..5. -.5 -. -.5 2 3 4 5 6 7 8 9 2 3 4 5 6
Repetitive Signals and Windowing The Fourier Transform is Defined over Infinite Time. In practice only a finite duration signal is available. (we can t wait forever!) Repetitive Signals -Take a snap shot of the signal Use an Integer Number of Cycles. -. 5 5 2 25 3 35 4 45 5 Ti m e
FFT Frequency Resolution Time Domain Frequency Dom ain Resolution Cycle Windows Resolution = /T 5 5 2 25 25Hz 75Hz Frequency (Hz) Time Domain 2 Cycle Windows Resolution = /2T Frequency Domain Resolution 25 5 75 25 5 75 2 225 25 37.5Hz 62.5Hz Frequency (Hz)
Cycle Windows and Interharmonics Time Domain Frequency Domain Resolution Harmonic 5 5 2 25 3 35 4 45 5 55 6 65 7 75 8 85 9 95 Frequency (Hz) 5 5 2 25 Interharmonics
Smearing or Gibb s Phenomenon. -. 5 5 2 25 3 35 4 45 5 Ti m e Smeared Spectrum Normalised Modulus Value.2.8.6.4.2.5.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5.5.5 2 Harm onic Num ber Synchronise the Samples to the Window!
Shaped Windows. -. 5 5 2 25 3 35 4 45 5 Tim e Smeared Spectrum.2.8.6.4.2 Normalised Modulus Value Amplitude.5.5.5.5 2 2.5 2.5 3 3.5 3.5 4 4.5 4.5 5 5.5 5.5 6 6.5 6.5 7 7.5 7.5 8 8.5 8.5 9 9.5 9.5.5.5.5.5 2 2 Harmonic Number
Non-Repetitive Signals Made of H, H2 and H3 But My Next Talk..
A.M. and Sidebands Carrier Modulator Time Domain Frequency Domain
Only Certain Frequencies Can Be seen
Picket Fencing: 5 Hz + 25 Hz Two Cycle FT.2 Amplitude.5.5 -.5 - -.5 64 28 92 256 32 384 448 52 Time Normalised Amplitude 2 3 4 5 6 7 8 9 Fourier Frequency 5Hz 25Hz One Cycle FT.2 Amplitude.5.5 -.5 - -.5 64 28 92 256 32 384 448 52 Time Normalised Amplitude 2 3 4 5 6 7 8 9 Fourier Frequency 5Hz 25Hz
Summary A Sampled Data System Sync Sample Timing Sample Clock Signal Anti-Aliasing Filter Analog to Digital Converter Transformed Data Data Storage and Processing