Where we’ve been Attenuate, Amplify, Linearize, Filter

Timing limitations to making measurements Gain-Bandwidth Product, etc. Thermal Mass, For Example

Where we’re going Speed, Storage Issues Frequency Space

Some signals are like this... time voltage

Some signals are like this... time voltage

But many signals are like this... time voltage

Introducing imaginary numbers ? ? ?

Introducing “ i ” (sometimes called ‘ j ’) Any number that is a multiple of i is an imaginary number:

Multiplication rules A real number times a real number is a real number A real number times an imaginary number is an imaginary number An imaginary number times an imaginary number is a real number

Powers of i anything to the “0” power = 1 anything to the “1” power = itself

Introducing complex numbers Any number that is a sum of a real number and an imaginary number is a complex number: imaginary part real part NOT “ 2i ”

Real versus imaginary parts All of the real components of a complex number, taken together, are the real part. The same holds for the imaginary part: imaginary part real part

Adding/subtracting complex numbers When adding (or subtracting) complex numbers, add (or subtract) the real and imaginary parts separately:

Multiplying complex numbers When multiplying a complex number by any other number, multiply both the real and imaginary parts

Multiplying complex numbers (continued) When multiplying two complex numbers, an easy method is the “FOIL” method: FirstOuterInnerLast

Example Perform the following addition. Identify the real and complex parts of the answer:

Example Perform the following subtraction. Identify the real and complex parts of the answer:

Example Perform the following multiplication. Identify the real and complex parts of the answer:

Complex numbers as vectors real axis imaginary axis complex plane

Magnitude of a complex number

Example Sketch the following complex numbers as vectors. What are their magnitudes?

Direction of a complex number A

Example What is the phase angle of each of these complex numbers?

Definition of the complex exponential Re Im

Magnitude of the complex exponential Re Im For any  1 1

Magnitude of the complex exponential Re Im Amplitude A A Simple harmonic motion! A·cos(135) The real part is what we observe.

 =  t Ae i  t is in effect a spinning complex vector that generates - a cosine function on the real axis and - a sine function on the imaginary axis re im x=Acos(  t) y=Asin(  t) F(t)= Ae i  t = A[cos(  t)+ i sin(  t)] = A  cos(  t) + iA  sin(  t) A A A A A A  t=0  t=90 0  t=180 0 Real part describes motion of mass on spring

Using a complex amplitude Re Im

Same magnitude, different phase Re Im t = 0 positions ( Ae i  t = e 0 = 1) B A

Using a complex amplitude Re Im Phase difference is still the same since both vectors rotated by 90 0 !

Real amplitude yields pure cosine wave in real space

Imaginary amplitude yields pure sine wave in real space

Complex Real amplitude = sine/cosine mixture

Sine waves can be mixed with DC signals, or with other sine waves to produce new waveforms. Here is one example of a complex waveform: V(t) = A o + A 1 sin  1 t + A 2 sin  2 t + A 3 sin  3 t + … + A n sin  n t --- in this case--- V(t) = A o + A 1 sin  1 t Ao Ao A1 A1 Fourier Analysis Just an AC component superimposed on a DC component

More dramatic results are obtained by mixing a sine wave of a particular frequency with exact multiples of the same frequency. We are adding harmonics to the fundamental frequency. For example, take the fundamental frequency and add 3rd harmonic (3 times the fundamental frequency) at reduced amplitude, and subsequently add its 5th, 7th and 9th harmonics: Fourier Analysis, cont’d the waveform begins to look more and more like a square wave.

This result illustrates a general principle first formulated by the French mathematician Joseph Fourier, namely that any complex waveform can be built up from a pure sine waves plus particular harmonics of the fundamental frequency. Square waves, triangular waves and sawtooth waves can all be produced in this way. (try plotting this using Excel) Fourier Analysis, cont’d