SEISMIC WAVE PROPAGATION Lecture 2: Fourier Analysis
Fourier Series & Fourier Transforms Fourier Series Review of trigonometric identities Analysing the square wave Fourier Transform Transforms of some common functions The delta function Sampling theorem and Nyquist frequency 2
Fourier Series * Joseph Fourier (768-83) For any function f(x) with period 2π, f(x) = f(2π +x), we can describe the f(x) in terms of an infinite sum of sines and cosines The general Fourier series may be written as: cos nx and sin nx are periodic on the interval 2π for any integer n. The a n and b n coefficients measure the strength of contribution from each harmonic. 3
Fourier Series To find the coefficients a, b n and a n, we multiply above equation by cos nx or sin nx and integrate it over interval -π<x<π. By the orthogonality relations of sin and cos functions, we can get a n = π b n = π a = π π π f(x) cos nxdx π π f(x) sin nxdx π π f(x) dx π π cos mx sin nxdx = for all m,n π π cos mx cos nxdx = m n = 2π m = n = = π m = n > π π sin mx sin nxdx = m n = π m = n > 4
Fourier Series An Example: The Square Wave f(x) = x < π = - π x < f(x) = f(x + 2 π) f(t) -6-5 -4-3 -2-2 3 4 5 a dt a n b n cos ntdt sin nt n n,2, 2 / n n,3,5, sin ntdt cos nt (cos n ) n n n 2,4,6, 5
Fourier Series An Example: The Square Wave 2 f ( t) sin t sin 3t sin 5t 2 3 5 f(t) -6-5 -4-3 -2-2 3 4 5 a dt a n b n cos ntdt sin nt n n,2, 2 / n n,3,5, sin ntdt cos nt (cos n ) n n n 2,4,6, 6
Fourier Series An Example: The Square Wave 2 f ( t) sin t sin 3t sin 5t 2 3 5 f(t) -6-5 -4-3 -2-2 3 4 5.5.5 -.5 7
Fourier Series The complex form of Fourier series 8
Fourier Transform The extension of a Fourier series for a non-periodic function; a generalization of the complex Fourier series in the limit as T Fourier series becomes an integral A waveform can be decomposed in terms of complex exponentials rather than sines and cosines The Fourier transform allows you to write any function, f(t), as the integral (sum) across frequencies of complex exponentials of different amplitudes and phases, F(ω) used in an enormous range of pure and applied science, including information processing, electronics and communications 9
Fourier Transform A signal has one or more frequencies in it, and can be viewed from two different standpoints: Time domain and Frequency domain Time domain: how a signal changes over time Frequency domain: how much of the signal lies within each given freq. band over a range of frequencies Fourier transform is the tool to connect the time domain and frequency domain. Why would we do the exchange between time domain and frequency domain?
Fourier space Fourier Transform Phase and amplitude spectrum Amplitude spectrum Phase spectrum The spectrum consists of two real-valued functions of angular frequency, the amplitude spectrum mod, A(), and the phase spectrum, f() F( ) A( ) i e ( ) In many cases the amplitude spectrum is the most important part to be considered. However there are cases where the phase spectrum plays an important role (resonance, seismometer)
Fourier Transform Spectral synthesis The red trace is the sum of all blue traces! 2
Fourier Transform Fourier Spectra: Main Cases random signals Random signals may contain all frequencies. A spectrum with constant contribution of all frequencies is called a white spectrum 3
Fourier Transform Fourier Spectra: Main Cases Gaussian signals The spectrum of a Gaussian function will itself be a Gaussian function. How does the spectrum change, if I make the Gaussian narrower and narrower? 4
Fourier Transform Fourier Spectra: Main Cases Transient waveform A transient waveform is a waveform limited in time (or space) in comparison with a harmonic wave form that is infinite 5
Fourier Transform Discrete Fourier Transform (DFT) Whatever we do on the computer with data will be based on the discrete Fourier transform F f k k N N j N j F e j f j e 2ikj / N 2ikj / N, k,,..., N, k,,..., N FFT is an efficient algorithm for computing the DFT 6 6
The delta function t ( t) and ( t) dt t Dirac comb Also called impulse function or a spike. t The act of digitizing a continuous signal into discrete values can be viewed as a multiplication of a string of delta functions spaced Δt apart 7
Sampling theorem and Nyquist frequency Discrete data How a time-domain waveform looks like? We need enough information in the samples to reconstruct the original waveform Nyquist criterion sample a sinusoid at least twice per cycle in order to know its true frequency. f s =/T 2 f max 8
Sampling theorem and Nyquist frequency An undersampled signal Important part of survey design and can affect survey costs and quality. It is important to sample signal correctly, but it is equally vital to adequately sample noise if this is to be removed by processing routines. Undersampling - suggests a different frequency of sampled signal 9