Lab 7 Fourier analysis and synthesis Fourier series

Lab 7: Fourier analysis and synthesis • Fourier series (periodic phenomena) • Fourier transform (aperiodic phenomena) • Fast Fourier transform (FFT) A powerful analytic tool that has many applications…. The Fourier Transform and its Applications Brad G. Osgood Stanford http: //see. stanford. edu/see/courseinfo. aspx? coll=84 d 174 c 2 -d 74 f-493 d-92 ae-c 3 f 45 c 0 ee 091

Applications of Fourier analysis Periodic phenomena (in space and time) • Physics Ø harmonic oscillation Ø waves (sounds, lights, and etc. ) • Acoustics • Image processing • Crystallography • Astronomy and earth science.

Fourier series (formula) A periodic function f(t) with period T: Any periodic function f(t) with period T can be mathematically expressed as a sum of harmonics.

Fourier series (animation) By Lucas V. Barbosa

Fourier analysis and synthesis Fourier analysis (Fourier transform) Given a periodic function f(t), calculate An and Bn In practice, f(t) is a waveform from measurement Fourier synthesis: Given An and Bn, reconstruct function f(t). Synthesize (arbitrary) periodic waveforms.

Symmetry of trigonometric functions odd even if f(t) is an even function if f(t) is an odd function,

Fourier analysis (an example) f Square wave 1 -T/2 t -1 Since f(t) is an odd function: Only need to calculate:

Fourier analysis (another example) Triangular wave Since f(t) is an even function: Only need to calculate:

Fourier synthesis Sum of Fourier components (e. g. square wave): “ 1+3”: “ 1+3+5+7”: “ 1+3+5+7+9”:

Fourier synthesis: square wave

Lab 7 B: Sawtooth wave: Odd function An=0 • Complete the derivation in your lab report!

Using complex exponential functions Using Eular formula, we could rewrite Sine and Cosine functions as complex exponential functions, which greatly simply the notation and algebra.

Reciprocal relationship and the conjugate variables Angular frequency Time domain Frequency domain

Fourier Transform (FT) A generalization of Fourier series to analyze aperiodic functions. E. g. random noise. Roughly speaking, the fundamental frequency tends to zero, i. e. 0, or T . Therefore, we need “All” frequencies (i. e. continuum limit) to describe aperiodic functions. In other words, the Fourier coefficients “become” a continuous function, and the Fourier sum becomes an integral, i. e. Fourier integral (inverse FT). Note: FT is also applicable for periodic functions.

FT and inverse FT Fourier Transform: Inverse FT:

Application of Fourier transform: find out periodic signal in a noisy background 0. 333 Hz

How does a spectrum analyzer perform Fourier transform? Sampling + Fast Fourier Transform (FFT) Most physical signals are continuous function of time. However, computers can only process discrete signal. To utilize the powerful computation capacity of modern computer, we need to covert a continuous signal to a digital signal, i. e. sampling. The hardware for sampling a voltage signal: analog-to-digital convertor (ADC)

Digitizing continuous signal (sampling) In signal processing, sampling is the reduction of a continuous signal to a discrete signal. A common example is the conversion of a sound wave (a continuous signal) to a sequence of samples (a discrete-time signal). http: //en. wikipedia. org/wiki/Signal_processing

An example: cell phone/radio http: //en. wikipedia. org/wiki/Signal_processing

Fast Fourier Transform (FFT) FFT was “invented” by Cooley and Tukey in 1965. It is a revolutionary numerical method that allows rapid and accurate Fourier analysis of discrete (digitized) signals. It is the fundamental mechanism behind any spectrum analyzer and many other digital (numerical) processing. Discrete FT ADC N points FFT N/2 points

Frequency span and resolution Spectrum analyzer can ONLY record finite number of data points. Same for the FFT spectrum (SR 760 records 400 points for the whole frequency span. The frequency bin size (resolution) = Freq Span/400, which is inversely proportional to total acquisition time (length of signal). Reciprocal relationship