Advantages And Disadvantages Of Decomposition

Question No # 2

Why do we decompose signal? Write some advantages of different types of decompositions.
The goal of decomposition is to replace a complicated problem with several easy ones. If the decomposition does not simplify the situation in some way, then nothing has been gained.

There are different ways to decompose signal in signal processing
• Impulse Decomposition
• Even Odd decomposition
• Unit step decomposition
• Interlaced decomposition
• Sinusoidal decomposition

 Impulse Decomposition:
Impulse decomposition breaks an N samples signal into N component signals, each containing N samples. Each of the component signals contains one point from the original signal, with the remainder of the values being zero. A single nonzero point in a string of zeros is called an impulse.
Advantage:
Impulse decomposition is important because it allows signals to be examined one sample at a time. Similarly, systems are characterized by how they respond to impulses. By knowing how a system responds to an impulse, the system's output can be calculated for any given input.

 Unit Step Decomposition :

Each component signal is a step, that is, the first samples have a value of zero, while the last samples are some constant value. Consider the decomposition of an N point signal, x[n], into the components: x0[n], x1[n], x2[n], …, xN-1[n]. The kth component signal, xk[n], is composed of zeros for points 0 through k - 1, while the remaining points have a value of: x[k] - x[k-1]. For example, the 5th component signal, x5[n], is composed of zeros for points 0 through 4, while the remaining samples have a value of: x[5] - x[4] (the difference between sample 4 and 5 of the original signal). As a special case, x0[n] has all of its samples equal to