Techniques in Signal and Data Processing CSC 508 Frequency Domain Analysis

In this section we learn how to transform a time domain signal into the frequency domain. We study both periodic and non-periodic signals and the types of transforms that work best with each. First Principles - We have combined sine waves of different amplitudes and frequencies to make other wave shapes. A sine wave is a periodic function described by the y-value of a unit-vector rotated about the origin. In this case the independent variable is the angle between the unit vector and the positive x-axis. We can specify a time-varying periodic function by defining a rotation rate  if the unit- vector in radians per second. The angle  of the unit vector is given by  =  t for time t.

There are three parameters that describe a sine function. These are the amplitude A, the frequency 2  and the phase  0. The amplitude is the peak value obtained by multiplying the value of the sine function by A. The frequency is the number of cycles of the sine wave generated per second and is given in 2  times  cycles per second (Hertz). The period (or wavelength) is the inverse of the frequency. The phase  0 is the amount of shif in angle (or time) of the wave relative to the origin. For a positive phase shift, the wave appears to move backqard because the values of the wave are reached earlier by an amount proportional to the phase angle. So what's the big deal about sine waves? They are only one of an infinitely large number of possible wave shapes. The reason we are concerned with sine waves is that they are a class of waves that can be combined to make any other wave shape. amplitude period = 1/frequency

Combining Sine Waves - Let's add two sine waves together whose amplitudes and frequencies are the same but whose phases are different by  /2. We see that this results in a new sine wave with the same frequency, an amplitude that is  2 times the original functions' amplitudes, and a phase that is half way between 0 and  /2. We can verify this result mathematically. If we were to reduce the amplitude of one of the sine waves in this example, the result would still be a sine wave but the phase of the resulting wave would shift toward the phase of the sine wave with the larger amplitude.

In order to build functions with other shapes we have to combine waves of different frequencies. In the next example with add two sine waves, buth this time we keep the amplitudes and phases equal and give one wave double the frequency of the other. Since the two wave have the same phase, they initially add together constructively (i.e. the amplitude of the resulting wave is greater). However, the higher frequency wave begins to swing back to negative values before the lower frequency wave and the two wave begin to cancel each other out (i.e. they add destructively). You may want to use the program sine_gen.exe to experiement with this effect.

The Fourier Series We are now ready to formalize the notion of a trignonometric series that can generate any function. Consider the following inifinite series, where 1/2 a 0 is a normalized coefficient establishing the baseline or neutral position of the periodic function being represented in the series. This series is called the Fourier series, named after its discoverer. The next pair of terms (a 1 cosx + b 1 sinx) is the fundamental, the next term is called the first harmonic or first overtone and so on. Since the cosine and sine are related by a phase angle p, this series can also be represented with only sine or cosine terms. For example, This is all very exciting, but to be of much value to us, we need to know how to determine the proper coefficients a n and b n to represent an arbitrary function f(x) in a Fourier series.

Suppose a function f(x) is represented by the Fourier series, We wish to solve for an expression of the coefficients a n and b n. To do this we must select one of the term in the infinite series at a time while forcing the other terms to zero. There is usually a trick in mathematical derivations and this case is no exception. Multiply both sides of this expression by cos mx and then integrate both sides with respect to x over one period (-  to  ).

The integrals in the summation can be evaluated with the help of a few trigonometric identities. In particular we have the desired selector operation, And the other integra is always zero. Therefore, for each value of the index m, only the mth term is non-zero, and we have, Solving for the coefficients a n and b n we have,

Any function f(x) that is continuous and integrable over the domain -  to  can be represented as a trigonometric series with the coefficients a n and b n, When f(x) is an even function (i.e. f(-x)=f(x) ) or an odd function (i.e. f(-x)=-f(x) ) then we can simplify the trigonometric series to a Fourier cosine series or a Fourier sine series respectively.  f(x) Evenf(x) Odd

Let's compute the coefficients of the square wave defined by,  We notice that the function is odd since f(-x)=-f(x), so we will compute the coefficients of the Fourier sine series. The limits of integration are between 0 and p so only the +1 portion of the square wave is used. (Don't forget that the -1 portion of the square wave is implied by the fact that we are assuming an odd function.) 0     cosine

So the coefficients are, etc. Using just these terms we get the approximation to the square wave shown below:

Homework: 1. Compute the first 8 coefficients of the following function and use sine_gen.exe to view the resulting approximation of f(x).             2. Define the limits of integration for this odd function and derive the formula for the coefficients of its Fourier sine series.