module 4: spectrum and spectrogram of signals · the main goal of this module is to learn how to...

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Summer 2019 CATALYST Research Project – Bits over the Air Module 4: Spectrum and Spectrogram of Signals © 2019 Christoph Studer ([email protected]); Version 0.3 The main goal of this module is to learn how to analyze the frequency components of sampled audio signals in MATLAB. First, you will learn that sampled signals can be represented in the so-called frequency domain. Second, you will see that one can simultaneously analyze the time-frequency properties of signals using a so-called spectrogram. Both of these tools will be useful in designing and understanding wireless communication systems. Remember: Whenever you are stuck, have questions, or are interested in learning more details about a specific aspect, please feel free to ask us—we are here to help!! 6 Frequency-Domain Representation of Signals So far, you have learned how to sample signals and how to visualize such signals as a function of time, which is called the time-domain representation of signals. In this module, you will learn that one can use MATLAB to represent signals also in the frequency domain. The frequency-domain representation reveals which frequencies are present and how strong they are in a given signal. As you will see, communication systems transmit information over a given frequency (or a frequency band, which is a set of neighboring frequencies). We will illustrate this important concept by analyzing the spectrum of a range of audio signals. The frequency-domain representation is another way of looking at signals, in which they are viewed as a function of frequency (and not of time). In addition to this, you will also learn that it is often useful to visualize both the time and frequency domains of the same signal; this can be accomplished by visualizing so-called spectrograms that show which frequencies are present in a signal at what time instant. You have probably seen such spectrograms before as they are very similar to these well-known “frequency analyzers” in 80s or 90s boomboxes—one of the goals is to understand what they actually show. 6.1 The Spectrum of a Signal Let us first inspect the spectrum of a very simple signal: a sine function at 440 Hz. Instead of using MATLAB functions to generate such a sine wave, we can also load one from the examples folder. Use the following command [y,FS] = load_audio('examples/sine-440Hz.wav'); to load a 5 second 440 Hz sine wave that was sampled at f s = 44, 100 Hz. Now, as in the previous module, plot the time-domain representation of this signal. Simply use plot_signal(y,FS) to look at the sample waveform in the time domain—you may have to zoom in to see the actual sine wave. To listen to this waveform, type 25

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Page 1: Module 4: Spectrum and Spectrogram of Signals · The main goal of this module is to learn how to analyze the frequency components of sampled audio signals in MATLAB. First, you will

Summer 2019 CATALYST Research Project – Bits over the Air

Module 4: Spectrum and Spectrogram of Signals

© 2019 Christoph Studer ([email protected]); Version 0.3

The main goal of this module is to learn how to analyze the frequency components of sampled audiosignals in MATLAB. First, you will learn that sampled signals can be represented in the so-called frequencydomain. Second, you will see that one can simultaneously analyze the time-frequency properties ofsignals using a so-called spectrogram. Both of these tools will be useful in designing and understandingwireless communication systems. Remember: Whenever you are stuck, have questions, or are interestedin learning more details about a specific aspect, please feel free to ask us—we are here to help!!

6 Frequency-Domain Representation of Signals

So far, you have learned how to sample signals and how to visualize such signals as a function of time,which is called the time-domain representation of signals. In this module, you will learn that one can useMATLAB to represent signals also in the frequency domain. The frequency-domain representation revealswhich frequencies are present and how strong they are in a given signal. As you will see, communicationsystems transmit information over a given frequency (or a frequency band, which is a set of neighboringfrequencies). We will illustrate this important concept by analyzing the spectrum of a range of audiosignals. The frequency-domain representation is another way of looking at signals, in which they are viewedas a function of frequency (and not of time). In addition to this, you will also learn that it is often useful tovisualize both the time and frequency domains of the same signal; this can be accomplished by visualizingso-called spectrograms that show which frequencies are present in a signal at what time instant. You haveprobably seen such spectrograms before as they are very similar to these well-known “frequency analyzers”in 80s or 90s boomboxes—one of the goals is to understand what they actually show.

6.1 The Spectrum of a Signal

Let us first inspect the spectrum of a very simple signal: a sine function at 440 Hz. Instead of usingMATLAB functions to generate such a sine wave, we can also load one from the examples folder. Use thefollowing command

[y,FS] = load_audio('examples/sine-440Hz.wav');

to load a 5 second 440 Hz sine wave that was sampled at fs = 44, 100 Hz. Now, as in the previous module,plot the time-domain representation of this signal. Simply use

plot_signal(y,FS)

to look at the sample waveform in the time domain—you may have to zoom in to see the actual sine wave.To listen to this waveform, type

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Page 2: Module 4: Spectrum and Spectrogram of Signals · The main goal of this module is to learn how to analyze the frequency components of sampled audio signals in MATLAB. First, you will

Summer 2019 CATALYST Research Project – Bits over the Air

play_audio(y,FS,OutID);

In case you forgot the output ID of your sound card, re-run the command get_audio_info. For thisspecific signal, we know that it only contains a single frequency at 440 Hz. It is interesting to note thatevery possible signal can be represented as a superposition of one or many sine and cosine waves atdifferent frequencies.

Assume for a moment that you have a continuous signal that is represented as a function f (t) wherethe time t is in the interval from 0 to 1 second. One of the central results in signal processing is that everypossible such function can be represented as an infinite series (or sum) of sine and cosine waves at differentfrequencies:

f (t) =a0

2+

∑n=1

(an cos(2πnt) + bn sin(2πnt)) . (4)

Here, the so-called Fourier coefficients (named after Jean-Baptiste Joseph Fourier, the inventor of thisimportant result) an and bn with n ∈ {0, 1, 2, . . .} characterize which frequency is represented how strong inthe function f (t). If you are not familiar with infinite series or the summation formula ∑, do not worry—itis not important to understand the mathematics behind this (you would learn this anyway in one of thefirst years while studying Electrical and Computer Engineering). If you are interested, feel free to ask ushow the coefficients an and bn can be calculated.

What is more important is to understand the intuition behind all this. Figure 8 provides an example. Onthe left figure, you can see a time-domain signal (think about it as an arbitrary function f (t) for t ∈ [0, 1]).On the right figure, we plot the amplitude of the coefficients an and bn for n = 1, 2, . . . , 8. This figure showswhich frequencies are present and how strong they are. You can see, for example, that at frequency indexn = 2 we have an = bn = 1/3. This means that the frequency with index n = 2 contains both a sine andcosine component which both have an amplitude of 1/3. While the concept of frequency index is a bitabstract (and we will not go into more details here), one can also visualize the frequency componentsdirectly in Hz but then we need to know with which frequency we sampled the signal of interest. Hence,we will do that on our 440 Hz sine waveform that we stored in the variable y.

time (seconds)0 0.2 0.4 0.6 0.8 1

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(b) Spectrum of time-domain signal.

Figure 8: Example of a time-domain signal and its frequency spectrum. The frequency spectrum showswhich frequencies are present in a time-domain signal and how strong these frequencies are.

To gain some more intuition, it is interesting to show the same signal as above but by not using alleight frequencies. Figure 9 shows what happens if we sum only one frequency, two frequencies, three

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Summer 2019 CATALYST Research Project – Bits over the Air

frequencies, etc. You can see that by superpositioning more and more sinusoids, we can approximate thefunction of interest. Mathematically speaking, we are taking the Fourier series in Equation (4) and do notsum to infinity but rather sum to 1, and then to 2, and then to 3, etc. Since the signal in Figure 8 containedonly 8 different frequencies, we can perfectly approximate this signal with a superposition of 8 sinusoids;this is what is shown in the last figure of Figure 9.

time (seconds)0 0.2 0.4 0.6 0.8 1

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(h) 8 coefficients.

Figure 9: Adding more and more sinusoids at different frequencies can approximate any function.

While the mathematics behind all this can be quite complicated, we can use MATLAB to do the hardwork for us. Run the following MATLAB command:

plot_spectrum(y,FS);

You should see a new plot that shows the frequency in Hz on the x-axis and the magnitude on the y-axis;this is the frequency-domain representation of the signal stored in the vector y. As expected, only onefrequency is present: 440 Hz. You can zoom in to confirm this fact. Note that all the other frequencies arevery close to zero. This spectrum plot implies that the signal in the frequency domain consists almostexclusively of a single frequency at 440 Hz. Furthermore, you can see that the range of frequencies goesfrom 0 Hz to 22,050 Hz, which is the highest frequency you can represent when sampling a signal atfs = 44, 100 Hz. Remember that such plots show the spectrum of a sampled signal.1

Activity 14: Visualize the spectrum of other signals

The folder examples contains other signals. Plot the spectrum of the wav-file sine-880Hz.wav.Can you see which frequencies are present in that signal? Also try the signal sine-1760Hz.wav.Note that it is always a good idea to listen to the waveform as well, to develop intuition.

Also plot the spectrum of the wav-file noise-white.wav. Then, try noise-pink.wav andnoise-brown.wav. What do you observe? Do these signals all contain the same frequencies?

1In case you are wondering how such plots are generated, the main technique is called the fast Fourier transform (FFT). Thistransform takes in any sampled signal and computes its spectrum. The FFT is probably the most widely used algorithm on thisplanet. Every cell-phone and laptop computes thousands of FFTs per second when transmitting data wirelessly!

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Summer 2019 CATALYST Research Project – Bits over the Air

6.2 The Spectrogram of a Signal

While analyzing the spectrum of signals is interesting and extremely important in the design of wirelesscommunication systems, it has one key limitation. To illustrate this limitation, load the following waveform

[y,FS] = load_audio('examples/chirp-100Hz-to-10000Hz.wav');

and listen to it. This wav-file contains a sinusoid that continuously changes its frequency from 100 Hz to10,000 Hz over time, starting with the low frequency and progressing to higher frequencies over 5 seconds.Such signals are called “chirps” and can be used to measure communication channels (or measure theacoustics of a given room). Now, plot the spectrum of this signal by typing:

plot_spectrum(y,FS);

As you will see, this audio signal contains frequencies from about 100 Hz to 10, 000 Hz as the filenamesuggests. While the spectrum shows that these frequencies are present during the 5 second duration ofthe signal, it does not show that the frequencies actually change over time. Put simply, the spectrum tellsyou which frequencies are present but does not tell you anything about its evolution over time. Hence, itwould be great to have a tool that visualizes both frequency and time—the spectrogram does exactly this!

Figure 10: Example spectrogram of a 5 second chirp signal that sweeps a single sine wave from 100 Hz to10, 000 Hz. Spectrograms show what frequency components are present at what time instant.

Let us first look at a spectrogram before we explain how they are generated. In MATLAB, simplyexecute the following command (which was provided in the zip-file you downloaded)

plot_spectrogram(y,FS);

to plot the spectrogram of the chirp signal that is stored in the variable y. You should see a figure that lookslike the one in Figure 10. The x-axis shows the time in seconds; the y-axis shows the frequency componentsranging from 0 Hz to fs/2 Hz. The colors encode the power (or intensity) measured in something thatis called decibels (dB, for short). (You can ask one of us if you are interested in learning more about

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Summer 2019 CATALYST Research Project – Bits over the Air

decibels, but basically it refers to the signal’s magnitudes measured on a logarithmic scale.) Brighter colorsindicate that certain frequencies are more strongly represented; darker colors imply certain frequencies areabsent; think of these as the magnitudes of the the Fourier coefficients an and bn. As you can see, there isa single frequency that starts at around 100 Hz and linearly increases towards 10, 000 Hz. In words, thespectrogram shows at what time which frequencies are contained in the signal.

Figure 11 illustrates the principle of spectrogram computation. The left side shows that one takesthe entire sampled signal and divides it into smaller blocks. For each smaller block, one computes thespectrum (this is called a short-time spectrum). One then generates a two-dimensional figure (as inFigure 10) where the columns correspond to a spectrum per block. The right side shows a typical spectrumanalyzer computer plug-in. These plug-ins (inspired by 80s and 90s boomboxes) simply visualize thefrequency spectrum for a small set of frequencies for separate blocks of the audio signal. Most of the time,the visualization is in sync with the audio signal, which allows one to see which frequencies are present atwhat time. The spectrogram as in Figure 10 is a static version of exactly the same idea.

(a) Principle of spectrogram calculation. (b) Typical spectrum analyzer.

Figure 11: Basics of spectrogram calculation. The signal is divided into small blocks for which the spectrumis calculated. Spectrum analyzer plug-ins (frequency equalizers) visualize the spectrogram over time.

Activity 15: Visualize the spectrogram of other signals

The folder examples contains a wide range of signals, including music and test sounds. Plot thespectrogram of these files and see whether you can observe in the spectrogram what you hear.

Activity 16: What is the highest frequency you can hear?

Visualize the spectrogram of the signal in the file chirp-100Hz-to-22050Hz.wav, which containsa chirp that goes to very high frequencies. Can you hear all frequencies up to 22, 050 Hz? Atwhat frequency can you not hear the signal anymore? Design an experiment to identify thefrequency that you cannot hear anymore. To this end, synthetically generate different sinewaves at different frequencies and play them back—you have learned in previous modules howto do that! At one point you should not be able to hear them anymore... What is that frequency?Important: Please be very cautious when playing test signals at high frequencies. Do not turnup the volume if you cannot hear a signal anymore! Note that even if you cannot hear signalsat high frequencies, they are still present and reaching your ears. Also, maybe some otherpeople in the room can hear them—also, dogs can hear them (that is how dog-whistles work).

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Summer 2019 CATALYST Research Project – Bits over the Air

6.3 Why is the Spectrum of a Signal Important?

The concept of frequencies and spectrum is central in the design of wireless communication systems.Different communication systems typically operate at different frequencies. For example, FM (shortfor frequency-modulation) radio uses frequencies in the electromagnetic spectrum from about 88 MHz(mega-Hertz, which stands for 1, 000, 000 Hz) to 108 MHz. Wireless LAN (also known as Wi-Fi or WLan)typically operates at frequencies of about 2.4 GHz (giga-Hertz, which stands for 1, 000, 000, 000 Hz) or evenhigher (e.g., 5 GHz). Many other communications standards for consumer electronics, such as Bluethooth(which are used for wireless headphones), also operate at around 2.4 GHz. All these communicationsystems (and many other systems used for television, emergency radio, defense, satellite communication,astronomy, weather radar, etc.) are operating simultaneously and they are occupying different frequencies.The are sharing the frequency spectrum.

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19

.8 19

.99

20.01

21

.0 21

.45

21.85

21

.924

22.0

22.85

5 23

.0 23

.2 23

.35

24.89

24

.99

25.01

25

.07

25.21

25

.33

25.55

25

.67

26.1

26.17

5 26

.48

26.95

26

.96

27.23

27

.41

27.54

28

.0 29

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.89

29.91

30

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BROA

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25 M

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LAN

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300

325

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525

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1900

20

00

2065

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94

2495

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30

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30 MHz 300 MHz

FIXE

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MOB

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MOB

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MOB

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LAND

MOB

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LAND

MOB

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BROADCASTING(TELEVISION)

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AERONAUTICALRADIONAVIGATION

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Mobile

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MET.

SATE

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MET.

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MET.

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MOB

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FIXED

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MAR

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FIXE

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(R)

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FIXE

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ATEU

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Land m

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Fixe

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30.0

30.56

32

.0 33

.0 34

.0 35

.0 36

.0 37

.0 37

.5 38

.0 38

.25

39.0

40.0

42.0

43.69

46

.6 47

.0 49

.6 50

.0 54

.0 72

.0 73

.0 74

.6 74

.8 75

.2 75

.4 76

.0 88

.0 10

8.0

117.9

75

121.9

375

123.0

875

123.5

875

128.8

125

132.0

125

136.0

13

7.0

137.0

25

137.1

75

137.8

25

138.0

14

4.0

146.0

14

8.0

149.9

15

0.05

150.8

15

2.855

15

4.0

156.2

475

156.7

625

156.8

375

157.0

375

157.1

875

157.4

5 16

1.575

16

1.625

16

1.775

16

1.962

5 16

1.987

5 16

2.012

5 16

3.037

5 17

3.2

173.4

17

4.0

216.0

21

7.0

219.0

22

0.0

222.0

22

5.0

300.0

FIXE

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Fixe

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LAND

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300.0

32

8.6

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39

9.9

400.0

5 40

0.15

401.0

40

2.0

403.0

40

6.0

406.1

41

0.0

420.0

45

0.0

454.0

45

5.0

456.0

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0.0

462.5

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462.7

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467.5

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467.7

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470.0

51

2.0

608.0

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4.0

698.0

76

3.0

775.0

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3.0

805.0

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6.0

809.0

84

9.0

851.0

85

4.0

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6.0

901.0

90

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9.0

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15.0

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00.0

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90.0

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95.0

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27.0

1429

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30.0

1432

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35.0

1525

.0 15

59.0

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10.6

1613

.8 16

26.5

1660

.0 16

60.5

1668

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70.0

1675

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95.0

1710

.0 17

61.0

1780

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50.0

2000

.0 20

20.0

2025

.0 21

10.0

2180

.0 22

00.0

2290

.0 23

00.0

2305

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10.0

2320

.0 23

45.0

2360

.0 23

90.0

2395

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17.0

2450

.0 24

83.5

2495

.0 25

00.0

2655

.0 26

90.0

2700

.0 29

00.0

3000

.0

300 MHz

AERO

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D AERONAUTICALRADIONAVIGATION

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EARTHEXPLORATION-

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(active)

Space research(active)

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RADIO-LOCATION

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AERONAUTICALRADIO -

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Amateur

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MOBIL

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RADI

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Mobil

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(Earth-to-space)(space-to-space)

EARTHEXPLORATION-

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(space-to-Earth)(space-to-space)

EARTHEXPLORATION-

SATELLITE(space-to-Earth)(space-to-space)

SPACEOPERATION

(space-to-Earth)(space-to-space)

MOBILE(ling of sight only

including aeronauticaltelemetry, but excludingflight testing of manned

aircraft)

FIXED (line of sight only)

FIXE

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Fixe

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(spa

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*

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*FI

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Earth exploration-satellite

(passive)

Space research(passive)

Radioastronomy

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FIXEDEARTH

EXPLORATION-SATELLITE

(passive)

RADIOASTRONOMY

SPAC

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Radiolocation

RADIOLOCATION

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Radio

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Fixe

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Space research(active)

Earthexploration-

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EARTHEXPLORATION-

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Fixe

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Radio

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Spac

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Space research(active)

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Space research(active)

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AERONAUTICAL

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Earthexploration-

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Earthexploration-

satellite (active)

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MARI

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3.0

3.1

3.3

3.5

3.6

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3.7

4.2

4.4

4.5

4.8

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4.99

5.0

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5.25

5.255

5.3

5 5.4

6 5.4

7 5.5

7 5.6

5.6

5 5.8

3 5.8

5 5.9

25

6.425

6.5

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6.7

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7.0

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7.1

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7.145

7.1

9 7.2

35

7.25

7.3

7.45

7.55

7.75

7.85

7.9

8.025

8.1

75

8.215

8.4

8.4

5 8.5

8.5

5 8.6

5 9.0

9.2

9.3

9.5

9.8

10

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10.5

10.55

10

.6 10

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10.7

11.7

12.2

12.7

13.25

13

.4 13

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14.0

14.2

14.4

14.5

14.71

45

14.8

15.13

65

15.35

15

.4 15

.43

15.63

15

.7 16

.6 17

.1 17

.2 17

.3 17

.7 17

.8 18

.3 18

.6 18

.8 19

.3 19

.7 20

.2 21

.2 21

.4 22

.0 22

.21

22.5

22.55

23

.55

23.6

24.0

24.05

24

.25

24.45

24

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24.75

25

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25.25

25

.5 27

.0 27

.5 29

.5 30

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ISM - 6.78 ± .015 MHz ISM - 13.560 ± .007 MHz ISM - 27.12 ± .163 MHz

ISM - 40.68 ± .02 MHz

3 GHzISM - 915.0± .13 MHz ISM - 2450.0± .50 MHz

3 GHz

ISM - 122.5± 0.500 GHz

This chart is a graphic single-point-in-time portrayal of the Table of Frequency Allocations used by the FCC and NTIA. As such, it may not completely reflect all aspects, i.e. footnotes and recent changes made to the Table of Frequency Allocations. Therefore, for complete information, users should consult the Table to determine the current status of U.S. allocations.

For sale by the Superintendent of Documents, U.S. Government Printing OfficeInternet: bookstore.gpo.gov Phone toll free (866) 512-1800; Washington, DC area (202) 512-1800

Facsimile: (202) 512-2250 Mail: Stop SSOP, Washington, DC 20402-0001

ISM - 61.25± 0.25 GHz ISM - 245.0± 1 GHz

AERONAUTICALMOBILE

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Figure 12: Frequency allocation chart of 2016 in the United States.

Figure 12 shows the United States frequency allocation chart from 2016 in the range of 1 kHz (kilo-Hertz, which stands for 1000 Hz) to 300 GHz. As you can see, nearly all frequencies are occupied bycommunication standards. Hence, when designing a communication system, one has to ensure to onlyoccupy the frequency range allocated for your transmission system. In fact, if you somehow causeinterference with neighboring systems, you will get into legal trouble with the FCC (and you have to payridiculously large fines!). To this end, when designing communication systems, one regularly uses theanalysis tools we have introduced before: the spectrum and the spectrogram.

One of the key goals of almost every communication system is to transmit as many bits per second(this is known as the throughput) as possible through the provided bandwidth. The bandwidth of acommunication system specifies how many contiguous frequencies one can use; a bandwidth of 1 MHzmeans, for example, that one can use the frequency spectrum from 20 MHz to 21 MHz. For this example,

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Page 7: Module 4: Spectrum and Spectrogram of Signals · The main goal of this module is to learn how to analyze the frequency components of sampled audio signals in MATLAB. First, you will

Summer 2019 CATALYST Research Project – Bits over the Air

the center frequency would be at 20.5 MHz (which is right in the middle of the two frequencies). Measuringthe amount of bits per second that can be transmitted per bandwidth is known as the spectral efficiency.Engineers who design communication systems are interested in improving spectral efficiency, rather thanachieving the highest possible throughput (also known as the data rate; the throughput is only a measure ofhow many bits per second we can reliably transmit). In contrast, marketing people are mainly interested inadvertising the highest possible throughput—this, however, does not tell you how many frequencies youare occupying (and cannot be used by other communication standards anymore). What most of us do notknow: It is very easy to design communication systems that achieve high throughputs (for example, onecan simply occupy all frequencies and not let anyone else transmit). It is, however, extremely difficult toachieve high throughput while only occupying a tiny fraction of the available frequency spectrum—that’swhy so many engineers are working on the design of wireless communication systems!

Activity 17: Compare spectral efficiencies of real-world communication systems

IEEE 802.11 is the name of many wireless LAN communication standards. If, for example, youare using Wi-Fi on your cell-phone, you are using one of the IEEE 802.11 standards. Since 1997,many wireless LAN standards have been proposed. The following table lists the key designparameter and performance metrics of some of them:

IEEE Standard Year Center Frequency [GHz] Bandwidth [MHz] Data rate [Mbit/s]

802.11g 2003 2.4 20 54802.11n 2009 2.4 40 600802.11ac 2013 5 160 3466.8802.11ay 2020 60 8,000 20,000

IEEE 802.11g, for example, has a spectral efficiency of 54/20 = 2.7 bit/s/Hz (simply take theratio of the data rate and the bandwidth; be careful with the units). Compute the spectralefficiency for the other three standards. What do you observe? Which of these standards isthe most efficient (in terms of spectral efficiency)? Which one achieves the highest data rate?Also, do you have an explanation why more recent standards are operating at higher centerfrequencies (up to 60 GHz for IEEE 802.11ay)? Let us know what you think!

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