Fourier Transform September 27, 2000 EE 64, Section 1 ©Michael R. Gustafson II Pratt School of Engineering

Recap of Chapter 3 Fourier Series Fourier Analysis Frequency-space representation of periodic signals Fourier Analysis Using the properties of the Fourier Series to solve system and circuit equations for periodic inputs Using the synthesis and analysis equations to examine frequency content of periodic signals

Limitation of Chapter 3 Fourier Series analysis only works for periodic signals!

Fourier Transform Another tool for solving system relationships in the frequency domain is the Fourier Transform The Fourier Transform can almost be thought of as the Fourier Series with an infinite period Huh?

Fourier Transform (cont) For the Fourier Series, discrete frequencies separated by the fundamental frequency are considered For the Fourier Transform, a continuum of frequencies separated by e are considered Because a continuum of frequencies are used, the signal does not have to be periodic

Motivation Remember this?

Motivation (cont) Nothing in that derivation presumed a periodic function, so the motivation is the same as for Fourier Series - "If it can be shown that a signal can be represented as a combination of complex exponentials, then the response of an LTI system to an input can be represented as a combination of complex exponentials times some function of the impulse response and the exponential."

Fourier Transform Equations Synthesis Analysis Note the factor is in front of the synthesis equation this time!

But what does THAT mean? For a signal, X(jw) gives the frequency content of the signal For a signal, the magnitude squared of the signal gives the power spectrum of the signal BUT, non-impulse values of X(jw) are only truly meaningful when looked at as a spectrum

Probability Analogy This is a probability density function of height (x-axis is difference from average)

Probability Analogy (cont) What is the probability of someone being exactly average height? What is the probability of someone being 1" taller than average? What is the probability of someone being 1" shorter than average? Don't those add up to over 100%???

Probability Analogy (cont) The probability density is only useful for determining ranges: What is the probability that someone is over a foot taller then average? What is the probability that someone is average height or shorter?

Back to X(jw) The analogy works here - while you cannot really say how much power is concentrated in a particular frequency, you can say how much is concentrated in a band of frequencies Exception->if the signal has periodic components, then the magnitudes squared impulses at those frequencies do represent the power there

But what about systems? If instead of x(t) you use h(t), H(jw) gives an indication of the input-to-output relationship of a system: The magnitude gives the ratio of the output to the input for a particular frequency The angle gives the phase difference between the output and the input Ex: if H(j10)=1.4<Pi/4, then the component of the output at 10 rad/s has 1.4 times the magnitude of the component of the input at that frequency and is 45 degrees ahead in phase

Convergence of the FT Same rules as FS except for periodicity:

Examples Find X(jw) for x(t)=u(t+W)-u(t-W) Find X(jw) for x(t)=exp(-at) u(t) Find x(t) if X(jw)=1 Find x(t) if X(jw)=u(w+W)-u(w-W) Find x(t) if X(jw)=2pd(w- w0)

Periodic Parts The Fourier Transform will only return finite values for energy signals. Power signals, if periodic, can be worked with:

Examples (cont) Find X(jw) for x(t)=cos(t)+u(t+1)-u(t-1) Find X(jw) for x(t)=cos(3t) sin(5t)

Questions ?