Short-Term Measurements

What is the point of this lecture?








Recursive
50 Hz LPF
Time-Domain Windowing
Let  denote a sequence to be analyzed. Let's limit the duration of  to  samples:

where  is a rectangular window and is defined as
.
The Fourier transform of  is given by:

The transform of  is given by:
.
This introduces frequency domain aliasing (the so-called picket fence effect):
Improvements Via Better Windows

Popular Windows
1. Rectangular:	
2. Generalized Hanning:	
	
	
3. Bartlett	
4. Kaiser	
5. Chebyshev:	
6. Gaussian	
There are many others. The most important characteristics are the width of the main lobe and the attenuation in the stop-band (height of highest sidelobe). The Hamming window is used quite extensively.
Recursive-in-Time Approaches
Define the short-term estimate of the power as:

We can view the above operation as a moving-average filter applied to the sequence .
This can be computed recursively using a linear constant-coefficient difference equation:

Common forms of this general equation are:
		(Leaky Integrator)
		(First-order weighted average)
		(2nd-order Integrator)
Of course, these are nothing more than various types of low-pass filters, or adaptive controllers. How do we compute the constants for these equations?

In what other applications have we seen such filters?
Relationship to Control Systems
The first-order systems can be related to physical quantities by observing that the system consists of one real pole:

 can be defined in terms of the bandwidth of the pole.
For second-order systems, we have a number of alternatives. Recall that a second-order system can consist of at most one zero and one pole and their complex conjugates. Classical filter design algorithms can be used to design the filter in terms of a bandwidth and an attenuation.
An alternate approach is to design the system in terms of its unit-step response:
There are many forms of such controllers (often known as servo-controllers). One very interesting family of such systems are those that correct to the velocity and acceleration of the input. All such systems can be implemented as a digital filter.
Time/Frequency Properties of Windows