Engineers need to know Fourier analysis

Fourier transform

Transcript

1 Fourier transformation short version 2nd semester Prof. Dr. Karlheinz Blankenbach University of Pforzheim, Tiefenbronner Str Pforzheim Overview / Applications / Motivation: The Fourier Transformation (FT) is used for the frequency analysis of (time) signals (signal processing), the filtering and the analysis of vibrations. The FT is also the basis for speech recognition. In FT, the Fourier amplitude is shown as a function of frequency, so you get information about which frequency is represented how strongly in the time signal. The associated algorithm (numerics) is called Fast Fourier Transformation (FFT). Time signal source: WIKIPEDIA Frequency range using FT To try out: GOOGLE PLAY: SimpleFFT, bs-spectrum, MS EXCEL Recommended literature: - Böhme: Analysis 2, Springer - Latussek et al. : Text and exercise book Mathematics V, Fachbuchverlag Leipzig-Cologne - Papula: Mathematics for engineers and natural scientists, Volume 2, Vieweg (only Fourier series!) - Burg et al. : Higher Mathematics for Engineers, Volume III, Teubner - Tilman Butz: Fourier Transformation for Pedestrians, Teubner Blankenbach / SS2013 /

2 Idealized example from music Fourier analysis of musical instruments How can one differentiate musical instruments when they all play the same frequency (here f o, e.g. 440 Hz)? Since this is known to be possible, the instruments must also emit additional frequencies, here harmonics with typically multiples of the basic frequency f o. Rel. Volume trumpet rel. Volume horn f o 2f o 3f o 4f o 5f o frequency f o 2f o 3f o 4f o 5f o frequency rel. Volume oboe rel. Clarinet volume fo 2f o 3f o 4f o 5f o Frequency fo 2f o 3f o 4f o 5f o Frequency The intensities (here relative volume, from the maths Fourier amplitude) of the oscillation frequencies are characteristic of the respective musical instrument FT spectra are idealized considerations. With real measurement in the time domain and Fourier transformation, these peaks broaden. Blankenbach / SS2013 /

3 Basic idea of ​​the Fourier transformation Known: Numerical approximation of functions by series (eg polynomial): - ex 1 + x + x² + - sinx x + 1/6 x³ + The example of sin shows that this approximation is more likely for periodic functions is unsuitable. Hence Fourier's approach to develop periodic functions with the periodic functions sine and cosine: k 1 Fourier series: f (t) a a coskt b decreases, ok k k is the frequency factor of (= 2 f). So there are only integer multiples of the basic frequency. Example: saw number function (here only sine occurs, plot next slide): (ft) here: 1 k k1 2 (1) sin (1 k relative amplitude The amplitude at the respective frequencies k represent a hyperbola (y = 1 / x ) dar kt) Explicit description of the first terms of the Fourier series: k amplitude / 3 f (t) 2 sint - sin2t + 2/3 sin3t Here: = 1 (see above) basic frequency 1st harmonic 2nd harmonic (meaning) of the sawtooth Harmonics Harmonics Thus, the sawtooth function is successively approximated by sine with increasing frequency (numerics). To try out: App Fourier series (Blankenbach / SS2013 /

4 Fourier representation sawtooth y 4 sawtooth (not to scale) up to k = 1 to k = 2 to k = t Zero offset by EXCEL step size bk Fourier coefficients sawtooth (spectrum) 2 1.8 1.6 1.4 1 , 2 1 0.8 0.6 0.4 0, line diagram, as there are individual discrete 'x values', here kk The bks fall relatively slowly because the tips of the sawtooth have to be reproduced. For k = 0, b k = 0; This can be explained technically by the fact that the sawtooth signal does not contain a DC voltage component. The Fourier series provides the series expansion according to sine and cosine for mathematically known functions. However, this method does not work for signals recorded using measurement technology, since only AD values ​​and no function are available here. That is why the Fourier transformation is used in practice! Blankenbach / SS2013 /

5 Fourier Transformation Idea: Analysis of a time signal in the frequency domain (spectrum) Description: f (t) F () complex representation (e jt = cost jsint) Definition of the integral transformation from the time to the frequency domain Reverse transformation from the frequency to the time domain Fourier integral 1 jt jt F () f (t) e dt f (t) F () ed 2 F () is Fourier transform of f (t): Spectral representation is generally complex, i.e. amplitude + phase ATTENTION: - Never = 2 Using / T is variable here. - Simplification for real even or odd functions f (t): see properties of FT # 9 Splitting F () into real and imaginary part e -jt = cost - jsint (Euler): F () jt f (t) e dt f (t) cos (t) dt jf (t) sin (t) dt F () R () j I () F () R² () I ² (): amount () IR () (): phase A () = F (): amplitude spectrum: practice! Blankenbach / SS2013 /

6 In practice: finished algorithm e.g. Butterfly (is not described here, as it is mostly implemented) for 2 n measuring points (512, 1024, ..). Blankenbach / SS2013 /

7 Further aspects FT of a square pulse: sinx / x FT of a square pulse: F () ~ sinx / x Representation often as magnitude sinx / x 1st secondary maximum (side lobe) derivative (sinx / x) = 0 (maximum) at = 3 / T with 5% of the maximum at zero (DC component) In technology often as a magnitude spectrum F () ² with scaling in decibels db = 10 log 10 (x) for voltage etc. with log (1) = 0 (f = / 2) location. 1. Sidelobe 9 / T over derivative (slope zero): Insert: F () ² = A² T² sin² (t / 2) / (T / 2) ² with = 9 / TF () ² = T² sin² (9/2 ) / (9/2) ² 0.05 T² where for = 0: F () ² = T² 1st side lobe approx. 5% of the maximum 10 log 10 (0.05) = -13 db power: db = 20 log 10 (x): Half power: -3 db = 20 log (0.5) Blankenbach / SS2013 /

8 examples of square-wave signals vs. optics (diffraction) Blankenbach / SS2013 /

9 Table Fourier transforms (from Föllinger, HÜTHIG) Compare rectangular pulse and sinx / x (Si) Blankenbach / SS2013 /

10 Blankenbach / SS2013 /

11 Blankenbach / SS2013 /

12 Java app for the Fourier transformer: Blankenbach / SS2013 /

13 Fourier transforms and window functions Procedure: Acquisition (e.g. osci) and multiplication in the time domain with window function Window functions dampen the secondary lobes (frequency does not exist in the original!) At the expense of the amplitude of the main maximum (the reason is a finite measuring time) Blankenbach / SS2013 /

14 Other window functions (from Butz: FT for pedestrians, Teubner) Blankenbach / SS2013 /

15 Frequency resolution of various window functions (from Butz: FT for pedestrians, Teubner) The following function is given: f (t) = cos (t) cos (1.15 t) cos (1.25 t) cos (2 t) cos ( 2.75 t) cos (3t) Frequency 1 1.15 1.25 2 2.75 3 Amplitude Question: Which window function is used to resolve the signal with neighboring frequencies and sometimes with low amplitudes? Blankenbach / SS2013 /

16 Fourier window function: Rectangular slit function (zoom, see below) Broadening of the 10 Hz peak F (amplitude spectrum) 1.2 Fourier transform of a time-limited cosine oscillation fo = 10 Hz, measurement duration 1s: 10 measured oscillations 1 0.8 0.6 0.4 0. f / Hz F (amplitude spectrum) Fourier transform of a temporally limited cosine oscillation fo = 10 Hz, measuring duration 10s: 100 measured oscillations f / Hz Side lobe damping due to more periods (longer measuring time), but risk of undersampling (too few time measured values per period). Blankenbach / SS2013 /

17 Example: Fourier transformation of an RLC resonant circuit with weak damping amplitude Damped oscillations 1 envelope 0, time -0.5-1 weakly damped Creep case Aperiodic borderline case rel. Amplitude 10 FT damped oscillation 8 6 A (d = 0.1) A (d = 0.25) A (d = 1), 5 1 1.5 2 2.5 rel. Frequency (w / w s) Blankenbach / SS2013 /

18 Exercises Fourier Transformation 1. Calculate the Fourier transform of a triangular pulse and sketch the result f (t) A - 0 T meas / 2 t Solution: 8A T m (F) sin² T ² 2 m 2. Calculate the FT of the double square pulse and sketch it Result f (t) 1-3 T -T 0 T 3T t Solution: F () 4 sint cos2 T 3. Perform the Fourier transformation for sin (628 t) with MS EXCEL and MATLAB. For further tasks see old exams. Blankenbach / SS2013 /