Table of contents | |
Introduction | |
Fourier Transform | |
Countable Infinity | |
Dot Product (Inner Product) of Vectors | |
Eigenvalue and Eigensignal | |
Conclusion |
A very basic concept in Signal and System analysis is the Transformation of signals. It involves a whole new paradigm of viewing signals in a context different from the natural domain of their occurrence.
Examples:
Example: Prove that the set of real numbers is not countably infinite.
- Suppose the set of real numbers is countably finite. Then every real number if mapped injectively onto the set of natural numbers.
- Let rk, where k N be the kth real number. Now we construct a real number r as follows: The integral part of r is 0.
- The kth decimal place of r is any integer that is different from the kth decimal place of rk.
- This number r which we have constructed differs from every rk at the kth decimal place. This contradicts our assumption that the set of real numbers is countably finite.
Note: A Discrete Signal x[n] can be thought of as a " Vector " with countably infinite dimensions. A Continuous Signal x(t) can be thought of as a vector with uncountably infinite dimensions.
Compare this with the definition of dot product for two finite-dimensional vectors. We will now introduce two new terms - "Eigenvalue" and "Eigensignal". These concepts will be used later along with the concept of inner product of signals to introduce the Fourier series.
"Eigen" is a German word meaning "one's own".
In the context of Signals & Systems, eigen signals and eigenvalues are described as follows:
In this lecture you have learnt:
41 videos|52 docs|33 tests
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1. What is the Fourier Transform? |
2. How does the Fourier Transform work? |
3. What is countable infinity in the context of signal transformation? |
4. What is the dot product (inner product) of vectors in signal transformation? |
5. What are eigenvalues and eigensignals in signal transformation? |
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