- Fourier Series
All periodic functions of frequency f can be expressed as a linear combination of sines and cosines of frequency f and multiples of f. Further, sines and cosines of a frequency f. and its multiples form a linearly independent set. That is, the set of sines and cosines of frequency f and its multiples form a basis for the vector space of periodic functions of frequency f.
Further, one can combine the sine and cosine functions of the frequency fto get a cosine (or a sine) of the frequency f with a phase shift (a shift from time t=0). The value of this shift depends on the relative value of the coefficients of the sine and cosine terms.
A function can be approximated using just a few cosine (or sine) terms, by discarding higher frequency terms. The more higher frequency terms are discarded, the more the distortion in the function, and the greater the compression.
See Fourier Series Approximation, with applet by Steve Crutchfield, update by Hsi Chen Lee. In this applet,
- For the rectangular pulse, view the quality of the approximation as the number of Fourier coefficients increases and decreases. Do this for other functions as well.
- Sketch a function of small frequency, with small-magnitude high-frequency fluctuations in the applet. Note the coefficients.
- Sketch a function of high frequency added to a small-magnitude low frequency term in the applet. Note the coefficients.
- Fourier Series and Harmonics
See Fourier Series Applet to hear the different frequency components.
- First, with the sound off, click on "sine". Notice that, in "sines" underneath, which notes the coefficients of the various sine functions that are combined to give the function on the screen, there is only one non-zero value. This makes sense. Now increase the zero coefficient next to this one with the cursor and watch the function change as the sine of twice the original frequency is added to it. When the cursor hovers over one of the two coefficients, you will see the contribution of the corresponding sinusoid on the screen.
- With the sound still off, click on the other functions. Notice the presence of harmonics (higher frequency multiples of the base frequency; i.e. more terms than a single one in the linear combination) in functions other than the sine and cosine, which are pure tones. Change the number of terms to see how good the approximations are, and how they improve as you increase the number of terms. Let the cursor hover about the "sines" or "cosines" to see the various contributions.
- Now switch the sound on. Notice the change in sound, due to the presence of harmonics, when the functions are of the same frequency but are not sine or cosine. Notice the coefficients of the sine and cosine and correlate with the sound.
- Similarly, look at : Listen to Fourier Series
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Image Processing:
Take a look at the effect of various image processing techniques on a single image, at Tutorial Images by Clifford Watson. Notice the following:
- Sampling rate: Consider the original image. It may be thought of as one represented using the standard basis, where the coefficients wrt the standard basis are the pixel values. The sampled image shown is an image constructed from every third pixel of the original. It is hence one where the basis is reduced to a third of the original one. The result is a smaller number of coefficients (compression) and a distorted image.
- Low Pass Filtering (skip quantization): This image has had a linear transform applied to it, consisting of a uniform blur of window size three by three. Compare it to the second low pass filtered image, created with a uniform blur of size five by five. Which image has better quality?
- Edge Detection: Both edge detection operations are also linear transforms.
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