Feb 1st

Basics of Signal Processing To make sure we all are on the same page, that is including those students who did not take Signals and Systems (or those who may not remember everything they learned there), we will spend some time on reviewing some very basic aspects of signal processing. The emphasis will be on basic principles – that is what is good to remember – rather than on mathematical derivations (you had or will have enough of it in your formal Signals and Systems). What we want to make sure is that all of us understand what we will be talking about later in this course, when we get to physiology and psychophysics of the perception and to applications of this knowledge.

ACTION describe waves in terms of their significant features SIGNAL SOURCE RECEIVER ACTION describe waves in terms of their significant features In analyzing any situation, that is when we take things apart and try to understand the individual elements, it matters what we are after (you remember the “Why?” question, right?). Let’s suppose we are facing the situation in this picture and we want to analyze the waves on the lake (you know where I am going – the waves could also represent the sensory signal). We can look on the waves ant try to describe them in some way, probably in terms of some of their important elements (features). This can be done without knowing where the waves came from, and where are they going. Similarly, the sensory signal can be analyzed as such. However, in many situations, it helps to know more about the situation. It may help to know how the waves originated since they may help us to describe them in more concise way. Similarly, sensory signals could come from reflections of light in some surfaces or from active light sources, or human speech comes from human vocal tract. This knowledge helps in guiding us what is it we may want to know about the signal. However, when you are on the boat, you may be mainly interested what the waves do to your boat, and be mulch less interested if the waves came from somebody throwing rocks in a water or from a wind. Similarly, when listening to speech, your main interest may be what you hear, may the speech come from a human being or from e.g. speaking bird or machine synthesizer. Finally, the man on the picture may be trying to communicate come message to the people on the boat. Then what will mainly matter is not how the waves look, how they were created, not even how they affect the boat, but mainly in the previously agreed upon code in recognized by the people on the boat. Similarly, in communication using sensory signals, it may mainly matter whether the message is being correctly decoded (recognized). Clearly, all these things,( i.e. the form of the signal, the way it was created, the way it is perceived, and what is the message the signal carries) interact and in most cases, it is good to understand all aspects of the situation. However, given our limited resources, we may want to focus more on one aspect or another – depending on what is it we need to know – WHY we do the analysis. So I like to differentiate among a) the signal-oriented analysis, b) the production-oriented analysis, c) the perception-oriented analysis, and d) the task-oriented analysis. It helps me to keep in mind what is it I should be after when doing the analysis job. understand the way the waves originate effect of the waves will the people in the boat notice ?

frequency = 1/T sine wave period (frequency) amplitude phase Imagine a string on a guitar. When picked (moved from its steady position), it starts vibrating. To derive its movement would take us a bit further we want to go here (any physics textbook can help you to understand it) but we know that the vibration is going to be periodic. For simplicity, lets assume no losses so the vibration is going to go forever. As the string moves, it compresses and expands molecules of the surrounding air (local air pressure changes) and these changes are propagating through space with speed of sound in the air. We have creating an acoustic wave with period T and wavelength lambda (these are related though the speech of sound as shown in the picture. You can imagine that at any point of space, the changes of the acoustic pressure will also change with period T. _ These changes in pressure can be converted to electric current through microphone (there is a number of microphone types, some are also shown in the picture). So at the output of the microphone we end up with electric signal that has sinusoidal shape. The sine wave is fully described by three parameters, 1) the period (or frequency-they are inverse of each other), 2) the amplitude, and 3) the phase (the phase is 0 in our picture, that means the signal starts at 0). frequency = 1/T sine wave period (frequency) amplitude phase l = speed of sound × T, where T is a period

sine cosine Phase F Here we have our sine wave again, this time also described mathematically using the SIN function (hence the constant PI that signifies the period of the SIN function). It is also shown that sine with 90 degree (PI/2) phase shift is called COS function. Also, the picture shows so called “unwrapped” phase which grows continuously with time (as opposed to so called “wrapped” phase which is always in the interval <0,2PI>.

Sinusoidal grating of image Acoustic signals are not the only signals that can be described by SIN function. Here we have an example of visual sinusoidal signal, where the intensity of light is changing sinusoidaly along the X axix. When such a picture is scanned as shown, the resulting vide signal will also be a sinusoid. Of course, here the period T is not related to speech of light but to the speed of the scanning  (or to spatial distance between identical intensities along the X axis – therefore this frequency is called a “spatial frequency”.

TO Fourier idea describe the signal by a sum of other well defined signals Some time ago (probably before 1822) Joseph Fourier got the idea of approximating periodic functions as an infinite sums of other (but much simpler) functions, namely by sinusoids. While it is true that not all periodic function can be approximated in the same way (mathematicians will tell that the function must obey so called Dirichlet conditions i.e. the function must have a finite number of extrema in any given interval, 2) must have a finite number of discontinuities in any given interval, 3) must be absolutely integrable over its period, and 4) must be bounded) it is good to remember that all periodic functions that describe real phenomena can be described by Fourier series.

Fourier Series A periodic function as an infinite weighted sum of simpler periodic functions! Here we have a mathematical form of Fourier series.

A good simple function Ti=T0 / i Here we have the functions that form the Fourier series. Here, notice the integer i that indicates the multiple of the fundamental frequency f0=1/T0, where T0 is fundamental period. That is, the Fourier series consist of harmonic functions (sine functions) with progressively increasing discrete set of frequencies which are all integral multiples if the fundamental frequency f0.

There is several alternative ways how to write the same thing There is several alternative ways how to write the same thing. You may safely ignore the last form (the one that uses complex exponential). I am trying to avoid use of complex numbers in this course but I am putting it here so that you know it exist.

e.t.c. ad infinitum Here comes the magic. We start with two sinusoids (both in phase, i.e. the phase is always 0), and keep adding (appropriately attenuated) sinusoids with progressively higher and higher frequencies, leaving every second out (i.e. adding only even integral multiples of the fundamental) until “infinity” (whatever that means in practical engineering), and end up with so called saw-tooth function. To get a real saw-tooth (with the leading edge perpendicular to the x-axis), we would really need an infinite number of sinusoids, since any sharp edge like we see in the saw-toots function requires truly infinite series.

T=1/f e.t.c…… T=1/f e.t.c…… Another function this time adding not only (again “appropriately attenuated”) even integral multiples of the fundamental but also including the odd multiples (thus using all higher harmonics). In this case we end up with so called square waveform.

Orthogonality Before we get to finding out which sinusoids to use in order to approximate any (reasonable –i.e. obeying Dirichlet conditions) periodic function, we need to understand so called orthogonality of functions. The periodic functions are said to be orthogonal if integral of their product over their period is always zero, unless these functions are identical. This property turns out to be very useful when we will try to figure out which functions to use in our (possibly infinite) Fourier series approximations.

+ - + - x + - x + - + = + - = area is positive (T/2) area is zero