In the previous session we discussed wavelet concepts like scaling and shifting
We will now look at two types of wavelet transforms
The continuous wavelet transform and the discrete wavelet transform
Key applications of continuous wavelet analysis are time frequency analysis and filtering of time localized frequency components
The key applications for discrete wavelet analysis are denoising and compression of signals and images
As I mentioned in the previous session these two transforms differ based on how they discretize the scale and the translation parameters
We will discuss these techniques as they apply in the 1d scenario
Let us take a closer look at the continuous wavelet transform or CWT
You can use this transform to obtain a simultaneous time frequency analysis of a signal
Analytic wavelets are best suited for time frequency analysis as these wavelets do not have negative frequency component
This list includes some of the analytic wavelets that are suitable for continuous wavelet analysis
The output of CWT are coefficients which are function of scale or frequency and time
Let us now discuss the process of constructing different wavelet scales
Recall from our previous video that when you scale a wavelet by a factor of 2
It results in reducing the equivalent frequency by an octave
With the CWT you have the added flexibility to analyze the signal at intermediary scales within each octave
This allows for fine scale analysis
This parameter is referred as the number of scales per octave
The higher the number of scales per octave, the finer the scale discretization
Typical values for this parameter are 10, 12, 16 and 32
The scales are multiplied with the sampling interval of the signal to obtain a physical significance
Here is an example of scales for a bump wavelet with 32 scales per octave
The signal is sampled every 7 microseconds
This is the corresponding plot with the equivalent frequency for the scales
Notice that the actual scale values are exponential
Now each scale wavelet is shifted in time along the entire length of the signal and compared with the original signal
You can repeat this process for all the scales resulting in coefficients that are a function of wavelet scale and shift parameter
To put it in perspective, a signal with 1000 samples analyzed with 20 scales results in 20000 coefficients
In this way you can better characterize oscillatory behavior in signals with the continuous wavelet transform
The discrete wavelet transform or DWT is ideal for denoising and compressing signals and images as it helps represent many naturally occurring signals and images with fewer coefficients
This enables a sparser representation
The base scale in DWT is set to 2
You can obtain different scales by raising this base scale to integers values represented in this way
The translation occurs at integer multiples represented in this equation
This process is often referred to as dyadic scaling and shifting
This kind of sampling eliminates redundancy in coefficients
The output of the transform yields the same number of coefficients as the length of the input signal
Therefore it requires less memory
The discrete wavelet transform process is equivalent to comparing a signal with discrete multi rate filter banks
Conceptually here is how it works
Given a signal s
The signal is first filtered with special low-pass and high-pass filters to yield low-pass and high-pass sub bands
We can refer to these as a1 and d1
Half of the samples are discarded after filtering as per the Nyquist criterion
The filters typically have a small number of coefficients and result in good computational performance
These filters also have the ability to reconstruct the sub bands while canceling any aliasing that occurs due to down sampling
For the next level of decomposition, the low-pass sub band a1 is iteratively filtered by the same technique to yield narrower sub bands a2 and d2 and so on
The length of the coefficients in each sub band is half of the number of coefficients in the preceding stage
With this technique, you can capture the signal of interest with a few large magnitude DWD coefficients while the noise in the signal results in smaller DWT coefficients
This way DWT helps analyze signals at progressively narrower sub bands at different resolution
It also helps denoise and compress signals