EC312 Lesson 14: Signals, Fourier and FiltersObjectives:(a) Understand the difference between the time domain and frequency domain representation for many different

signals.

(b) Recognize that all periodic signals can be described as a sum of sinusoidal frequencies (Fourier) (not responsible for equations) and be able to visualize the effects harmonics have in the time domain and frequency domain.

(c) Know the basic filter types and their characteristic frequency plots: bandpass, band-reject, low pass and high pass filters.

(d) Be able to analyze the RLC series circuit as a tuned circuit and describe its equations for resonance, cutoff frequencies, BW and Q, and be able to analyze the AC circuit for impedance and voltage divider output.

We have already introduced the frequency domain in the first and second lessons of the wireless section and we have seen the frequency vs time relationship. We understand that when we want to communicate wirelessly, we will need to modulate our baseband or information signal and that it is important to know the frequencies that are transmitted and the bandwidth. We have introduced the idea of filtering in the last lesson to describe how to tune in a band of frequencies of interest when receiving that signal.

We will now be concentrating on ideas to help us solve the frequency content of different signals in the time domain, and then we will delve into circuits which will be able to filter that signal in terms of allowing certain frequencies of the signal to pass and attenuate other frequencies. We will analyze the RLC series Bandpass circuit specifically, but you should understand that all of the types of filters could be analyzed using your EE331 skills, and that you are responsible for knowing the effects the four types of filters have on frequency.

So let’s review a bit and look at some of those original signals and then introduce more complicated ones.

This is a 440 Hz sinusoid: in the “time domain” which is what we are accustomed to, the signal looks like this:

0 0.005 0.01 0.015 0.02 0.025-0.15

-0.1

-0.05

0

0.05

0.1

0.15

Time (sec)

Vol

tage

(V)

time(sec)

We can also plot this signal in the “frequency domain”. The x-axis is frequency, and the y-axis is the strength or the magnitude of the signal at the particular frequency (it could be units of volts or decibels ).

0 500 1000 1500 2000 2500 3000 3500 40000

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Frequency (Hz)

Vol

tage

(V)

frequency (Hz)

f = 440 Hz

1

Now, consider the more complex signal shown below on the left. In the frequency domain, it looks like the figure on the right.

1 1.005 1.01 1.015 1.02 1.025-1

-0.5

0

0.5

1

Time (sec)

Vol

tage

(V)

0 500 1000 1500 2000 2500 3000 3500 40000

0.05

0.1

0.15

0.2

0.25

Frequency (Hz)

Vol

tage

(V)

time (sec) frequency (Hz)These signals were both periodic, what about non-periodic signals?

Most signals used in communications systems are complex and non-periodic.Consider the sample speech file.

0 0.5 1 1.5 2 2.5 3

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

Time (sec)

Vol

tage

(V)

2

The frequency content of the non-periodic portion of the signal displays a continuous band off frequencies.

3

To transmit these baseband signals, we have learned that we need to modulate them. The technique we have investigated is Amplitude Modulation, and we have seen the effects in time and frequency that this process represents. So let’s review it, to remind ourselves of the time-domain equations and to recall the frequency domain representation in terms of sidebands and bandwidth.

4

5

The frequency spectrum of an amplitude modulated signal with the baseband signal consisting of a single frequency, fm, shows the Bandwidth is twice the fm.

The frequency spectrum of an Amplitude Modulated signal with the baseband signal composed of many frequencies, shows that the Bandwidth is twice the highest frequency component.

6

For the non- periodic signals, the Bandwidth is also twice the highest frequency component.

So how do we know what frequencies are present in a signal?Any periodic function f(t) with period T can be represented by weighted sines and cosines.Specifically, any periodic function with period T can be represented in the frequency domain with specific amplitudes occurring only at specific frequencies.What are the specific frequencies?

Answer: The fundamental frequency and all the integer multiples of the fundamental frequency. These multiples of the fundamental frequency are called the harmonics.

So, a periodic signal with period 0.01 seconds will have amplitudes only at the frequencies of 100 Hz, 200 Hz, 300 Hz, etc. Some of these amplitudes might be zero. In this case, 100 Hz is the fundamental frequency, 200 Hz is the second harmonic, 300 Hz is the third harmonic, etc.

So, when a periodic signal is represented in the frequency domain, it can only have amplitudes at the fundamental and harmonic frequencies.

What are the specific amplitudes?

Answer: There is a mathematical technique called Fourier analysis. We will skip the details and just use the results.

01fT

7

The Square Wave

The Fourier series for a square wave is given by:

Note the following:

The series consists only of odd harmonics. The amplitude of the nth harmonic is (4V/π)*1/n. The more harmonics that are added, the closer the approximation comes to the perfect square wave.

Frequency domain representation:

Below are time-domain plots of Fourier series approximations using various numbers of harmonics.

4 1 1 3 1 5sin 2 sin 2 sin 2 ...3 5

Vf t t t tT T T

8

Square wave using 20 odd harmonics:

Demo links: http://www.falstad.com/fourier/ http://ptolemy.eecs.berkeley.edu/eecs20/week8/examples.htmlYou can also experiment with this using your MATLAB skills. The following link provides some code to generate a portion of the Fourier series to approximate a square wave: http://www.mathworks.com/products/matlab/examples.html?file=/products/demos/shipping/matlab/xfourier.html

Fourier Series and Filters Recall that filter circuits are designed to pass some frequencies and block others.

Consider the application of a square wave to a low-pass filter as shown below.

9

Bandwidth of a Signal Although many Fourier series representations contain an infinite number of harmonics, often only a few significant harmonics are required to obtain a satisfactory representation.

Bandwidth is a quantity of fundamental importance in communications systems. Bandwidth is one of the two principle limiting factors in the performance of communication systems

(the other limiting factor is noise) The “information capacity” that a signal can carry is directly proportional to its bandwidth.

Back to the radio! Consider tuning an FM radio station. Remember what allows your radio to isolate one station from all of the adjacent stations? Do you remember the bandpass filter?

Why do we need to restrict the bandwidth of channel? Think about the other stations.

So now that we can find the frequency content of our signal and we know the bandwidth we want, we need to design the filter circuit that does the job of tuning our station in.

The specific tuned circuit, which we will be investigating, is the passive RLC series circuit. The book describes another passive tuned filter, the parallel RLC circuit, and it describes active tuned circuits, which provide amplification as well as filtering, and it also describes passive low pass and high pass filter circuits. We will use

10

your background from EE331 to analyze just the series filter circuit and this will help us understand the concepts behind all of the circuits, but this is the only analysis you are responsible for.

The Series RLC Circuit

Suppose an AC circuit contains an inductor of value L.

What is the value of ?

What is the value of ?

As frequency, f, increases, what happens to ?

Sketch a picture of versus f. f

Suppose an AC circuit contains a capacitor of value C.

What is the value of ?

What is the value of ?

As frequency increases, what happens to ?

Sketch a picture of versus f. f

Consider an AC voltage source connected to an inductor, a capacitor, and a resistor all in series. A series RLC circuit is made up of inductance, capacitance, and resistance as shown below.

What is the impedance of the circuit as seen by the source?

1

L CR jX jXjR j LC

R j LC

Z

LX

LX

LZ

LX

LX

CX

CX

CZ

CX

CX

11

The magnitude of the impedance is:

When is the magnitude of Z minimized?

Impedance (Z) is minimized when the inductive and capacitive reactances are equal (XL = XC).

At what frequency f is this true?

Resonance

The impedance (Z) is minimized when the inductive and capacitive reactances are equal (XL = XC). The frequency where this occurs is called the resonant frequency (f r) and is given by:

Resonance is a balanced condition between inductive and capacitive reactance. At resonance, the series RLC impedance is purely resistive. All communications equipment contains tuned circuits like this one: circuits made of capacitors and inductors that resonate at specific frequencies.

2

22 2 1L CZ R X X R L

C

Z

1 12 02L CX X L fL

C fC

12rf LC

12rf LC

12

So what? Why do we care?

Let us consider the magnitude of the current I as a function of frequency.

Below is a plot of the current amplitude versus frequency.

So, a tuned circuit (like this) is frequency selective; that is, it has a strong response at the resonant frequency, and a range of frequencies around the resonant frequency (the bandwidth).

Half-power Frequencies

The average power dissipated by this circuit is: P = I 2

R = V 2/ Rwhich peaks at f = fr .

The frequencies f1 and f2 shown above are defined as the half-power or cutoff frequencies. The upper and lower cutoff frequencies f1 and f2 are often called the 3 dB frequencies because at these points the circuit’s power dissipation is down by a factor of one-half from its peak.

It can be shown that the resonant frequency is the geometric mean of the cutoff frequencies:

22 1

S V VIZ

R LC

VIZ

1 2rf f f

13

The bandwidth (BW) of the tuned circuit is defined as the difference between the upper and lower cutoff frequencies.

Bandwidth (BW)Quality Factor

The “sharpness” of the of the frequency response of a resonant circuit is measured by the quality factor, Q. The quality factor can be defined as the ratio of the resonant frequency to the bandwidth:

It can be shown for a series RLC circuit that:

2 1BW f f

rfQBW

14

For a high-Q circuit (Q 10), the upper and lower half-power frequencies (f1, f2) are approximately symmetric about fr so:

Example Problem 1 Calculate the impedance (Z) of the circuit below at f = 2 MHZ with the indicated component values L = 1 H

C = 0.01 F

Example Problem 2 Suppose you have a series RLC circuit with a 10 pF capacitor that you want to resonate at 40 MHz. What value of inductance should you use?

Example Problem 3

(a) What is the resonant frequency fr of the series RLC circuit below? (b) At this frequency, what is the impedance of the circuit?(c) At the resonant frequency, what is the phase difference between the voltage across the circuit and the current through it?

2Q =BW

r L rf X f LR R

1 2, ,2 2r r

BW BWf f f f

15

Example Problem 4 What is the bandwidth of a resonant circuit with a frequency of 20 MHz and a Q of 50?

16