Course Index

BM661 - Module 1: Introduction to Signals and Systems

Lecture 1A

Signals

A signal is any physical quantity that varies as a function of one or more independent variables.
In this course we shall be concerned mainly with 1D signals. The one dimension that we shall deal with will usually be time.
The principles used in the discussion of 1D signals can be extended to multidimensional signals.
Examples: Variation of aortic pressure, variation of muscle force
The signal is usually represented as, p(t), where "t" is the independent variable (usually time) and "p" is the dependent variable (the amplitude or strength of the signal).

1-Dimensional signals

Single independent variable - usually time.
Amplitude varies with time - amplitude is a function of time
Examples: Variation of aortic pressure, variation of muscle force
The signal is usually represented as, p(t), where "t" is the independent variable (usually time) and "p" is the dependent variable (the amplitude or strength of the signal).

2-Dimensional signals - flat pictures or images

Two independent variables - in the case of a picture, the two variables are along the length (X) and breadth (Y) of the image.
The light intensity, or colour varies with the position on the image - colour is a function of the X and Y variables.
Examples: (a) light intensity reflected from a black & white photograph, (b) the elevation of terrain above sea-level.
The signal is usually represented as, p(x,y), where "x" and "y" are the independent variables (along the length and breadth in the case of flat pictures) and "p" is the dependent variable (the amplitude or strength of the signal).

3-Dimensional signals

Three independent variables - in the case of a solid object, the three variables are along the length (X) and breadth (Y) and height (Z) of the object.
The density of the material varies with the position within the object - the mass density is a function of the X and Y variables.
A motion picture is also an example of a 3-D signal. The three independent variables are the screen length (X) and breadth (Y) and time (Z).

Examples of 1-D Time signals

One dimensional time signals are very commonly encountered in biomedical engineering. The following figure shows a "snapshot" of one cycle of a normal electrocardiogram.

But in contrast to the above "static" picture which is commonly used in texts, look at the following example which shows a more realistic representation of a time signal changing continually. The observation window shows a small section of the signal in the immediate past.

Interactive example: Time signals

Important operations on signals

The following are the most important operations on time signals.

Time shifting: A time signal can be delayed (to the past) or advanced (to the future). A given signal, x(t), can be time shifted by T to form x(t-T). If T is positive then the signal has been delayed and if T is negative it has been advanced.
Time scaling: A time signal can be speeded up or slowed down. A given signal, x(t), can be time scaled by a factor K to form x(Kt). If K<1 then the signal has been slowed down, if K>1 then it has been speeded up. A speeded up signal appears compressed and a slowed down signal appears expanded.
Time reversal: A time signal can be observed in a time-reversed fashion. In practice this can be achieved by recording a signal and then replaying it backwards. x(t) time reversed would be represented as x(-t).
In general all the above operations can be combined and applied to a particular time signal.

Use the accompanying program to learn more about the above operations on time signals. Use the program as follows:

The panel on the left shows a continuously presented time signal. You can choose one of several available choices.
In order to work on a small segment of the signal, capture a portion using "Hold" and then pressing the "Copy" button. The selected segment will be copied to the right panel.
Now use the controls on the left side to manipulate the selected signal. Compare the panels on the left and right to understand how the signal is being altered.
The three controls, shifting, scaling, reversal, yield the function: x(dKt-T); where 'd' is the time direction (+ forward, - reversed), 'K' is the scaling factor, 'T' is the time shift value.

Interactive example: Time signal operations

Try these:

Capture a segement of ECG. Notice that a "snapshot" does not always give the ECG waves in normal order.
Try combinations of time shift and time scaling. See how time scaling affects the time shift.
Try combinations of time shift and time reversal. See how time reversal affects the time shift.
Try combinations of all three, time shift, time scaling and time reversal. Write down the expressions for each resulting signal.
When the sinusoid is selected, the effect time shift and time reversal is difficult to notice. But nevertheless they are significant. If the frequency of the sinuosid were lower then the differences will be more noticeable.
Select a segment from the Aortic Pressure waveform. Using pencil and paper, copy the selected signal, x(t) and draw the signals corresponding to the following expressions: x(t/2 - 1), x(-t-3), x(-2t+3). Then check them against the program.