Fourier Series Fourier Transform Discrete Fourier Transform ISAT 300 Instrumentation and Measurement Spring 2000

Review of Notation â Imaginary Numbers The square root of -1 = i (or j if you are an engineer). Wheeler and Ganji use j â Complex Numbers A number that has a REAL part and an IMAGINARY part. Example: a + jb Magnitude of this complex number

Review of Notation â Where  is the angular frequency,  f t is time A is the amplitude (can be complex) â An oscillatory complex function

Fourier Series Analysis â Recall that Fourier Series Analysis is one common technique for the spectral analysis of time-varying signals. â Fourier Series Analysis tells us what component frequencies must be “added up” to get the signal being studied. â Stated differently, this analysis tells us the frequency content of a time-varying signal.

Some New Notation â It can be shown that âThese relationships can be used to write the Fourier series in a different way. âThe “Complex Exponential Form” of the Fourier Series.

Fourier Series -- Complex Exponential Form â Fourier Series for a continuous, periodic signal  o = fundamental or first harmonic angular frequency e jn   t represents the n th harmonic (the n th frequency component). c n is the amplitude of the n th frequency component. Note: this is an “infinite series”. (There may be a typographical error in your text.)

Fourier Series -- Complex Exponential Form âEach coefficient c n is, in general, a complex number. âThese coefficients are calculated by evaluating an integral:

The Fourier Transform âWhile the Fourier Series is the easiest way to begin to learn about spectral analysis, the Fourier Transform is the most commonly used spectral analysis technique. âThe Fourier Transform can be applied to any practical signal. âIt does not require that signal to be periodic.

The Fourier Transform  F(  ) is a continuous, complex-valued function.  The above equation transforms a function of “ t ” to a function of “  ”. âTransforms from the “time domain” to the “frequency domain”. âThe Fourier Transform of a signal (function) f(t) is defined as

The Inverse Fourier Transform  The above equation transforms a function of “  ” to a function of “ t ”. âTransforms from the “frequency domain” to the “time domain”.  If the Fourier Transform F(  ) is known, the original function can be recovered by calculating the Inverse Fourier Transform.

Why is he so sad? Jean Baptiste Joseph Fourier born on March 21, 1768

Discrete Fourier Transform (DFT) Calculation of the Fourier Transform of a signal involves some complicated math. But there is some good news. The computer can make this task easy. Recognize that, in an experiment, a signal is measured only over a finite (limited) time period. Also, with a computerized data acquisition system, the signal is measured only at discrete times  sampling.

Discrete Fourier Transform (DFT) For signal data taken at discrete times (sampled) over a finite (limited) time interval, the Discrete Fourier Transform (DFT) has been defined as follows: N is the number of samples taken during a time period T. The increment of f is called  f = 1/T. The increment of time between samples (the sampling period) is  t = T/N.

Discrete Fourier Transform (DFT) Here it is again: Note that there are N values of F. The F ’s are complex numbers. The F ’s are complex coefficients of a series of sinusoids with frequencies of 0,  f, 2  f, 3  f, …, (N-1)  f. The amplitude of F for a given frequency represents the contribution of that frequency to the original signal.

Discrete Fourier Transform (DFT) Here it is again: Since N samples of the original signal were taken in a total time T, the sampling rate = N/T.

Discrete Fourier Transform (DFT) Why is this important? To find the frequency content of a signal. To design an audio format (e.g., CD audio). To design a communications system (what bandwidth is required?). To determine the frequency response of a structure. A bridge. A building. A musical instrument.

Discrete Fourier Transform (DFT) Sound waves for A above middle C (440Hz)

Discrete Fourier Transform (DFT) Magnitudes of DFTs of sound waves.