(a) Use (9.142) to construct the "mother wavelet" corresponding to the hat function (9.140). (b)

step 1 We have three functions in complex space. Function 1 is 1. Function 2 is the sign of pi x, and f

(a) Use (9.142) to construct the "mother wavelet"...

Key Concepts

Orthogonality in Wavelets

Orthogonality in wavelet theory refers to the property that different wavelet functions, whether arising from scaling or as parts of the wavelet family, do not overlap in energy. This ensures that each component in a wavelet expansion is independent, which simplifies the analysis and reconstruction of signals. Orthogonality between the mother wavelet and the scaling function, as well as among different translates, is often a desirable property for creating efficient and non-redundant representations.

Integer Translates in Wavelet Analysis

Integer translates involve shifting a function by whole number amounts. In the context of wavelet analysis, showing that a function and its integer translates are orthogonal is crucial for forming a stable basis for signal representation. This property ensures that the contributions from different shifts do not interfere with each other, which is particularly important in discrete wavelet transformations and practical implementations.

Mother Wavelet

A mother wavelet is a single function used to generate a family of wavelets through dilations and translations. In wavelet analysis, this function captures the details or high-frequency components of signals. It is constructed so that its scaled and shifted versions create an orthogonal basis for L2(R), allowing for efficient signal decomposition and reconstruction.

Scaling Function

The scaling function, often associated with the low-pass filter in multi-resolution analysis, is used to approximate a function at a coarse resolution. In many constructions, a specific scaling function, such as the hat function, is used to build the wavelet basis via refinement equations. This function is central to defining the hierarchical structure of the approximation spaces in wavelet theory.

Transcript

00:01 We have three functions in complex space.

00:09 Function 1 is 1.

00:14 Function 2 is the sign of pi x, and function 3 is the cosine of pi x.

00:33 We need to show that they are orthogonal first.

00:38 So the inner product of f1, f2, would be the integral over the interval from negative 1 to 1 of f of x, f1 of x, which is just going to be the sign of pi x.

01:09 So we take the integral and we get the sign of pi x.

01:26 That should be cosine, opposite of cosine of pi x over pi, from negative 1 to 1.

01:40 So the cosine of pi is negative 1, and the cosine of the opposite of the opposite of pi is also negative 1.

01:53 So we get negative 1 over pi minus negative 1 over pi, which is 0.

02:10 F1, the inner product of f1 and f3, integral from negative 1 to 1 of the cosine of pi x, which would be the sign of pi x over pi x over pi, from negative 1 to 1.

02:33 The sign of pi is 0.

02:36 The sign of negative pi is 0.

02:38 That's going to give us 0 minus 0, which is 0.

02:47 F2, f3, the inner product.

02:49 That's going to be the integral from negative 1 to 1 of sine of pi x, cosine of pi.

03:06 X dx should have been writing dx on all these which is the same as the integral from negative one to one using double angle identities um two sine x cosine x equals a sign of two x so we could just write in sine of 2 pi x over 2 d x, which is going to be negative cosine of 2x over 2 pi x, which would be negative cosine of 2 pi x over 2 times 2 pi, which would be 4 pi, from negative 1 to 1 to 1.

04:28 So that's going to be the cosine of 2 pi, which is the same as the cosine of 0, and the cosine of 0 is 1.

04:38 And then the second one is going to be the cosine of negative 2 pi, which is going to be the same thing, 1.

04:43 So it's going to give us negative 1 over 4 pi minus negative 1 over 4 pi, which is 0.

04:57 Now we need to make an orthonormal set.

05:02 And so first thing we need to do is we need to do the norm of f1, which is going to be the square root of the integral from negative 1 to 1 of f1 of f1 is just 1.

05:28 F1 squared is just going to be 1 squared.

05:32 Dx.

05:34 Okay, so that's going to give us the integral of 1 is x from negative 1 to 1.

05:52 Okay.

05:53 Oh, good.

05:54 And that's going to give us the square root of 1 minus negative 1.

05:59 I thought we were going to end up with 0, which would be not good.

06:04 So that's the square root of 2...