Delay operator in feedfoward filters

[Whoohw]

Homework Statement

The symbol z = e^(iw) is called the delay operator. Explain why (in terms of the phasor) z^(-k) corresponds to a delay of k sampling intervals.

The Attempt at a Solution

We really have only discussed how to manipulate the filter equations, not really waht teh delay operator does.

I know a filter can be written as Y = b0*X + b1*X*Z^-1 + b2*X*Z^-2... etc

Where X is the original signal... This can also be written as
y[n] = b0*x[n] + b1*x[n-1] + b2*x[n-2]...etc

Where y[n] is the output signal and x[n] is the input signal

In the second form it is clear that x[n-1],x[n-2] is a delayed form of x[n], but i do not know how to explain why, in terms of the phasor, Z^-k corresponds ot the delay

 

[KataKoniK]

of k sampling intervals.

it is important to understand the fundamental concepts and principles behind the equations and symbols that we use in our work. In this case, the symbol z = e^(iw) is known as the delay operator because it represents a phase shift in the frequency domain. In other words, it describes how the signal changes over time.

When we raise z to a negative power, such as z^(-k), we are essentially reversing the direction of the phase shift. This means that the signal is moving backwards in time, or in other words, it is being delayed by k sampling intervals.

To understand this in terms of the phasor, we can think of z as a vector with a magnitude of 1 and a phase angle of w. When we raise z to a negative power, we are essentially multiplying the magnitude by 1 and dividing the phase angle by k. This results in a vector with the same magnitude, but a phase angle that is k times smaller.

In the frequency domain, a phase shift corresponds to a delay in time. So when we have a smaller phase angle, it means that the signal is delayed by a smaller amount of time. Therefore, raising z to a negative power corresponds to a delay of k sampling intervals because the phase angle is k times smaller, resulting in a smaller delay in time.

In summary, the delay operator z = e^(iw) represents a phase shift in the frequency domain, and raising it to a negative power corresponds to a reverse phase shift, or a delay in time. This can be understood in terms of the phasor by considering the changes in magnitude and phase angle.