As Haar is multi-resolution technique (MRA) for signal representation, i.e the window used in Haar is not constant unlike STFT. Hence it is very easy to identify the variation of signal's frequency at different time, for a non-stationary signal. Hence we would get a 2D representation of a 1D signal in transformed domain, as like continues wavelet transform, S-transform and STFT. Where one axis represents the time, another shows the frequency and third one illustrates the magnitude. It means all the MRA and STFT not only yields the frequency content of a signal but also emphasis at what time these frequencies occurred in transformed domain.
I understand the concept of Haar and other transformation quite a bit and also able to generate time-frequency joint representation of a signal for continuous wavelet transform. However, I am not able to understand how time-frequency joint representation can be possible for discrete Haar transform and also unable to represent it in MATLAB for a 1D time varying signal. Please suggest some paper or idea about the concept of Joint representation of a signal in discrete Haar transformed domain (Only for Discrete transform, not continuous).
Consider a time varying signal ( a0, a1, a2, ...... , am) where m=2^n. The high pass Haar filter is (1/2,1/2) and low pass filter is (1/2,-1/2). On applying Haar filters, we get two vectors each of length m/2 one high frequencies( b0, b1, ... , bm/2) and other low pass finer details (d0, d1, ...., dm/2) . Here bi = (a2i + a2i+1)/2 and di = (a2i - a2i+1)/2.
The values of a's can be obtained from bi and di as a2i = bi+di and a2i+1 = bi - di.
For example if signal is (12, 24, 32, 42, 10, 64, 22, 54) then its decomposition are
(18, 37, 37,38) and (-6, -5, -27, -16).
Should you need values on higher resolution then signal B can be further decomposed in higher and lower frequencies.
@R.C. Mittal ... Sir, i know the concept how to decompose the signal in two bands of Approximation and Detail coefficients. However i need to know the frequency components that the signal is carrying in the transformed domain for Harr Wavelet transform. For example if we have a sinusoidal signal of three different frequency components such as: 10Hz (For first 0 to 100 time samples) , 25Hz (From 101 to 200 time samples) and 50 Hz (From 201 to 300 time samples) then as per the concept of Multi-resolution analysis (MRA), we must get a 2d representation of this 1D signal in transformed domain, where X- axis will represent the frequency, Y-axis shows the time at which these frequencies are occurred, while the other axis yields the magnitude of that signal. I think this concept is valid for every transformation techniques that comes under MRA and also for STFT. However for Discrete wavelet transform we are getting same 1D signal output as shown in your example. If this is the scenario then how we can discernment the signals frequency in different times.
A very interesting and simple tutorial on wavelet:
http://person.hst.aau.dk/enk/ST8/wavelet_tutotial.pdf
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