Moulton  Lectures

 

On

Electro-Acoustics

 

 

 

Presented by:

Dave L Moulton

 

 

 

 

Location:  Thales Acoustics Harrow UK

Date:  12-March-2003

 

Content

 

I now want to look at a very important mathematical tool which will give an insight into the world of digital signal processing and digital noise cancellation.  That tool is known as the Z-transform

 

So far all of my lectures have involved analogue signals and the analogue processing of those signals, for example, acoustic noise (unwanted pressure wave) is an analogue signal.  All of the processing methods that I have taught so far have been analogue, such as acoustic and electrical  filters, acoustic and  electrical transfer functions.

 

I now want to use the fundamental theory of signal sampling as a way of introducing  you to the Z-transform, and a means of showing what the transforms represents and how it can be used. 

 

My next few lectures will be focused on Digital Noise Cancellation where the Z-transform will feature a lot in the mathematical representation of the systems.

 

 

On with the lecture

 

Analogue system Representation

Lets start by going back to basics and showing a very straight forward representation of an analogue system.

 

Figure 16.1

 

In this representation the input signal x(t) and the output signal y(t) are analogue, and they are both functions of time.   The output signal has had its amplitude response modified by the processing function h(t).  The method of modification could be a combination of differentiation, Integration and amplitude scaling.

 

Now consider the same system with the processing function h(t) set to a value of 1.  In this case the output is an exact representation of the input.

 

 

Figure 16.2

 

Sampled Representation

Now lets do something out of the ordinary and replace the processing function with a simple switch.

 

Figure 16.3

 

In this case it should be very obvious that we will only get an output y(t) = x(t) when the switch is closed. This will only occur when the switch contacts are physically touching each other, for all other positions of the switch the output y(t) = 0.

 

Now lets assume that we are able to operate the switch in such a way that we can open and close the switch contacts at a constant rate. If we let the period of opening and closing of the contacts (Time between each consecutive contact closure) be represented as T_s.  Then the frequency of the switch action can be defined as:

 

If we now let our processing function take the form of a continuously operating switch we will get an output that looks something like that shown in figure 16.4.

 

Figure 16.4

 

Now lets take a closer look at the discrete output.  Figure 16.5 shows a representation of the output signal.  Note how the output signal is broken into discrete samples of the input signal.  Each sample occurs precisely at the moment the two switch contacts touch each other. The amplitude value of the output is the value of the input function x(t) at each of the sampling times.

 

Figure 16.5

 

Mathematical Description of the Sampled Signal

In order to fully understand the sampling process we need to find a way of mathematically representing the sampled output signal.

 

Lets describe the un-sampled analogue output as y(t) and the sampled output signal as y^*(t). 

 

From Figure 16.5 we can see that the sampled output could be represented as a sequence of numbers thus, if we had N samples:

 

For example the sequence could be:

 

Y^*(t) ®10, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1……………..[16.3]

 

This representation shows a sequence of numbers all separated by commas, but it tells us very little about the action of the switch.  There is no information in this sequence of numbers about the rate at which the input data was sampled or the possible affects of the sampling rate. 

Using the Detla Function to Represent the Action of the Switch

By now you should be familiar with the delta function and its definition.

 

Looking back at figure 16.5 we can represent each switch action as a delta function occurring at that moment in time.  The scaling properties of the delta function allow us to multiply it with the value of the input function x(t) at each discrete sampling point.  We end up with a series of delta functions all scaled to the discrete value of the input function.

 

Figure 16.6

 

We can now represent our simple switch as a generator of a stream of impulses.  Thus the mathematical representation of the switch ‘Sw’ is:

 

Thus our input signal x(t) can be represented as a sampled output  y^*(t):

 

 

We now have a mathematical description of the sampled output data as a function of the discrete sampled values x(NT_s) and the switch action

d(t-NT_s).

 

The Laplace Transform of the Sampled Signal

So far we have created a time domain representation of our sampled signal y^*(t).  In order to progress further we need to know something about the frequency domain response of the sample signal, since the rate at which we sample our input signal x(t) could have an adverse affect on the purity of our original analogue signal. 

 

In this lecture I will not go into detail about the affect of the sampling frequency, however I will take the Laplace transform of the sampled signal and from that provide the definition of the Z-transform.

 

We Perform the Laplace transform in the usual way:

 

 

In this expression only the delta function is function of time t, the sampled input signal x(NT_s) is not a direct function of t.  Thus we can rewrite [16.6] as:

 

Applying an exponential shift to the delta function results in the Laplace Transform:

 

 

Definition of the Z-Transform

The Z-transform is defined mathematically as:

 

 

z  represents a single period time shift.  In this case the time shift is an advance of one sample period T_s, whereas z^-1 represents a time lag (delay) of one sample period. 

 

Note that the output function Y^*(s) now becomes Y(z), the sampled data stream.

 

Expanding this sequence gives:

 

 

We now have a means of converting a sequence of sampled data values into a more useable mathematical sequence. Thus for a switch capable of producing pure delta functions our sampled data shown in [16.3] can be expressed as:

 

Y^*(t) ®10, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1……………..[16.3]

 

 

 

The Zero Order Sample and Hold

In the real world it is not possible to get a switch to perform the action of a true delta sampler. This is simply because a delta function is an ideal concept and not a realizable function.  As you now know a true delta function has infinite magnitude and a width that is infinitely small.  The closest we can get to a delta function is a pulse with a very small but finite width.  In the real world there is another problem that we must overcome, which is the time taken to capture the value of each sample pulse. Most real world systems require a minimum hold time for each sample so that the capturing device, usually an Analogue to Digital Converter (  A to D) is able to correctly read the data value.  Clearly in the case of a true delta function sampler the sampled value is only available for an infinitely short duration and is not practical to capture.

Figure 16.7

 

A more realistic sampler is known as the zero order sample and hold. This type of sampler captures the signal and holds the value until the next sample occurs. The affect of a zero order sample and hold on our example input signal is shown below in figure 16.8.

 

Figure 16.8

 

The mathematical representation of this type of sampler can be established by considering that each pulse is basically made up of a combination of time shifted unit step functions.  For example lets look at how a single pulse is formed:

 

 

Figure 16.9

 

In the time domain we can describe a stream of sample pulses as:

 

 

Thus the sampled output signal y^*(t) will take the form:

 

 

As with the delta function sample, we can take the Laplace Transform for the zero order hold sampled data, thus

 

Again using an exponential shift on the Step Functions we get:

 

A little bit of simplification:

 

Leading to our Z-transform representation:

 

A little more manipulating

 

Combining  [16.8] and [16.9]

 

To keep everything in terms of z we simply write:

 

The large Z symbol means look up the z-transform of the term in brackets. There are many books of tables showing the z-transforms and inverse z-transforms of many different functions.

 

We can represent a First Order sample and Hold system by making a modification to our simple delta switch configuration shown in figure 16.4. Resulting in the configuration shown in figure 16.10.

 

Figure 16.10

 

Properties of the Z-transform

Like the Laplace Transform the transfer function is the ratio of the input to the output.

 

Figure 16.11

 

Likewise

 

 

 

 

End of Lecture