Lecture 4: Frequency and Spectrum THURSDAY, JANUARY 24, 2019

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1 Lecture 4: Frequency and Spectrum DANIEL WELLER THURSDAY, JANUARY 24, 2019

2 Agenda Frequency and periodicity Light and frequency Frequencies and harmonics in music Complex numbers Spectrum definition and the Fourier transform Frequency vs. time: the spectrogram Fortunately, Dr. Elizabeth was able to invert the transform. Image credit: Randall Munroe/xkcd.com 2

3 Frequency content of a signal A signal is a function of time x(t) in audio, or a function of space I(x,y) in an image. Another way to describe a signal is by the frequency content. Let us start by imagining a single sinusoid (a cosine function or a sine function, for example): This sinusoid has three parameters: the amplitude A, the frequency f, and the phase φ. The period is the smallest time T such that x(t) = x(t+t) for all t. What is the period of this sinusoid? 3

4 What is frequency? In the sinusoid example, the frequency f is the inverse of the period T. The faster the signal repeats, the higher its frequency. So the frequency measures how fast a single tone repeats. Yes, but it can be used in other contexts as well, even when the signal is not periodic What are the units of frequency? Generally speaking, frequency is measured in cycles/unit time. When time t is in seconds, frequency f is in cycles/second, or Hertz (Hz). Music generally measures notes relative to high A, which has a frequency of 440 Hz. How fast is 440 Hz? 4

What is frequency? We can also measure frequency in radians/unit time. This measure of frequency, ω = 2πf, is commonly encountered when describing the behavior of analog electronic circuits.

5 What is frequency? We can also measure frequency in radians/unit time. This measure of frequency, ω = 2πf, is commonly encountered when describing the behavior of analog electronic circuits. 440 Hz = radians/second We are used to describing many kinds of signals in terms of their frequency content. Light (frequency = color) Radio (frequency = station) Music (frequency = note) Video (frequency = frame rate) Heart beat (frequency = heart rate) Brain waves (frequency = alpha/beta/gamma/ ) An electrocardiogram (ECG or EKG) measures electrical impulses near the surface of the heart. Image credit: (left) Patrick J. Lynch, C. Carl Jaffe/Wikipedia. 5

6 Light and frequency Visible light exists in many colors. These colors reflect the frequency, or equivalently the wavelength, of the light wave: wavelength (λ) = speed of light (c) / frequency (ν) Violet light has the highest frequency ( THz), and the shortest wavelength ( nm). Next comes blue light ( THz), then green ( THz), yellow ( THz), orange ( THz), and finally red light ( THz). The energy (E) of the corresponding light photon increases with increasing frequency: h = Planck s constant ( J*s) 6

Cones are sensitive to particular ranges of frequencies corresponding roughly to blue, green, and red.

7 Light and frequency In fact, how we perceive visible light is due to its frequency: The back of the eye (retina) contains two types of light-sensitive cells: rods and cones. Rods are sensitive to a wide range of light frequencies. Cones are sensitive to particular ranges of frequencies corresponding roughly to blue, green, and red. The optic nerve transmits signals from these cells to the brain, which synthesizes them into colors. Image credit: Arizona Board of Regents/ASU Ask a Biologist. 7

Cameras and colors 1855: The idea of three color-sensitive cone cells was posited by James Clerk Maxwell (yes, that Maxwell) as a possible basis for color photography.

8 Cameras and colors 1855: The idea of three color-sensitive cone cells was posited by James Clerk Maxwell (yes, that Maxwell) as a possible basis for color photography. 1868: Louis du Hauron patented the basic process for producing color photographs on paper. This idea is based on subtractive color, i.e., forming colors by removing components of white light (that contains all colors). This is also how color printers work, with cyan, magenta, and yellow ink instead of red, green, and blue. 1935: Eastman Kodak introduced color camera film (Kodachrome), with three layers of emulsion that recorded the red, green, and blue components of light. Agfa developed improved film a year later, that would allow developing all three colors simultaneously. 1963: Polaroid introduced color film for their instant camera. 8

9 Cameras and colors Digital cameras work similarly: Silicon-based sensors convert incident photons into energy, read out as voltages. Filters placed on these sensors make them sensitive to specific ranges of light. Cameras can arrange these sensors in an alternating mosaic like the Bayer pattern on the right. Image demosaicking takes the red, green, and blue images and aligns them to form a single RGB color image. Some modern cameras can stack these sensors and filters to get red, green, and blue together. 9

Light and frequency Other electromagnetic radiation is also characterized by its frequency: Ultraviolet (7.5 10 14 3 10 16 Hz) Infrared (3 10 11 4.

10 Light and frequency Other electromagnetic radiation is also characterized by its frequency: Ultraviolet ( Hz) Infrared ( Hz) X-rays ( Hz) Gamma rays (> Hz) Microwaves ( Hz) Radio waves ( Hz) Energy increases as frequency increases this is why ultraviolet radiation can cause skin cancer, and repeated, significant X-ray or gamma radiation exposure can be deadly. Fortunately, extremely high energy radiation from the sun is mostly disrupted by the earth s magnetic field and the ozone layer. Or we d all be cooked like a rotisserie chicken! Image credit: Con Poulos/epicurious.com. 10

11 Sound waves and music Like light, sound propagates as a wave, and hence can be described as a sinusoid (or a collection of sinusoids). Humans can hear sounds with frequencies between ~20 20,000 Hz (this varies with age, and personto-person) Dogs can hear up to nearly 50,000 Hz! Dolphins can hear up to 150,000 Hz!!! Pure sinusoids are called tones. These have single frequencies. High A = 440 Hz Middle C = 262 Hz Each octave corresponds to a doubling in frequency. Actual musical notes are more complicated Image credit: NASA (public domain)/wikipedia. 11

12 Sound waves and music Music generally consists of chords that contain multiple notes simultaneously. Here are 2 tones: 12

13 Sound waves and music We can superimpose three tones to form a triad: 13

14 Sound waves and music Note the relationship between frequency and notes on the musical octave scale: one semitone frequency increases by 2 1/12 each octave has 12 semitones 2 1/12 = A B C D E F G A B C D E F 440 Hz 880 Hz one octave frequency doubles G A 1760 Hz 14

15 Sound waves and music 12 semitones = 1 octave 7 semitones = perfect fifth A C# E 4 semitones = major third A 440 Hz 440 x 2 4/12 = Hz 440 x 2 7/12 =659.3 Hz 440 x 2 12/12 = 880 Hz 15

16 Sound waves and music Pure tones or combinations of pure tones are fine, but they don t sound very musical. What are we missing? Harmonics! Musical instruments don t play pure tones. They play periodic signals that are shaped by the characteristic of the instrument. These periodic signals not only have a frequency spike at the fundamental frequency; they also have spikes at multiples of that frequency. These multiples, or overtones, correspond to the 2 nd n th harmonics of the instrument. Different instruments weight these harmonics relatively differently to produce different sound. 16

17 Piano A Instrument examples Trumpet A Piano Trumpet 17

18 Instrument examples We see the same periodicity (high A ) in the waveforms, but the character is different: Piano Trumpet 18

19 Musical instruments There s a lot more that goes into the shape of a musical waveform and its impact on the corresponding sound. We won t go into it in this class The envelope of the waveform: attack, sustain, and release The timbre of the instrument: the texture created by the envelope and harmonic weightings The underlying oscillation (cosine or some other waveform) Loudness or volume Tempo 19

20 Pure tones and frequency spikes Given a pure tone x(t) = cos(2πft), the frequency representation X(f) consists of a discontinuous spike at exactly ±f. More formally we call this spike a Dirac delta function δ(f-f 0 ), centered at frequency f 0. It has infinite height at frequency f 0, and is zero at all other frequencies. Aside: In a strict mathematical sense, this function is undefined at f 0. Instead, we define it implicitly, via the integral equation where X(f) is some real-valued function. Actually, even this definition is not strictly correct, but it s close enough. The proper definition would require a graduate course in mathematical analysis Setting X(f) = 1, we observe that the delta function has unit area. 20

21 Pure tones and frequency spikes Note, we haven t said anything about amplitude or phase here. This is because the spike s location depends only on the sinusoid s frequency. The amplitude A and phase φ will appear as a complex number Ae iφ multiplying the positive frequency (f 0 ) spike and Ae -iφ multiplying the negative (-f 0 ) spike. (A/2)e -iφ δ(f+f 0 ) X(f) (A/2)e iφ δ(f-f 0 ) -f 0 f 0 Frequency, f Aside: let s learn a little bit about complex numbers. 21

22 Aside: Complex numbers Complex numbers are composed of two separate components: a real number part an imaginary number part What is an imaginary number? It is the square root of a negative number. Recall positive numbers have positive and negative square roots (e.g., 2 2 = (-2) 2 = 4). Negative numbers have positive imaginary and negative imaginary square roots. (i 2 = (-i) 2 = -1) We can think of an imaginary number as being a real number, multiplied by the square root of -1 (called i). If b is real, then (bi) 2 = (-bi) 2 = -(b 2 ). Thus, we represent a complex number c as the sum of a real number a and an imaginary number bi: c = a + bi. Arithmetic is all performed as usual, using this summation form and distributing appropriately. 22

23 Aside: Complex numbers Example: add 1+2i and -3+4i Example: multiply -1-i and 1+i 23

24 Aside: Complex numbers In addition, we have the notion of a complex conjugate. For a complex number c = a + bi, its complex conjugate, called c*, is equal to a - bi. If we multiply c = a+bi by c*, we get the square of its magnitude, c 2 = a 2 + b 2. This allows us to simplify division: Example: divide 3-i by 2+2i 24

25 Aside: Complex numbers In addition to representing complex number c as a sum of real and imaginary parts a+bi, we can represent it in its polar form, in terms of its magnitude c and phase c. Note that the phase angle is between -π and π (or between 0 and 2π) radians, so we will need the signs of b and a to distinguish between tan -1 (b/a) and tan -1 (b/a)+π. Then, we have the polar form This polar form is related to Euler s relation: In fact, this relation allows us to go back to the summation form: 25

26 Aside: Complex numbers Example: Convert 2-2i to polar form Example: Convert 4e -iπ/3 to summation form 26

27 Aside: Complex numbers In general, a real-valued signal x(t) will have complex-valued frequency content X(f). We will see this more clearly once we define the spectrum formally. For a sinusoid, the magnitude of the complex number will tell us the amplitude, while the phase of the complex number will tell us the phase of the sinusoid. 27

28 Frequency content: the spectrum We have mainly viewed frequency content for pure tones, which can be described as spikes. Other periodic signals can be approximated as a weighted combination of sinusoids at different frequencies. This combination is called a Fourier series. We can take a finite number of these sinusoids and plot spikes for each of them, scaled by their relative weightings. The visualization or plot of the relative composition of frequency content for a signal is called its spectrum. We will see that most aperiodic signals have a spectrum as well. In general, a spectrum includes both positive and negative frequencies. We will see it is enough to describe real-valued signals using just the positive frequency axis. 28

29 The spectrum Fundamental Tenet I: There is a one-to-one relationship between a signal and its frequency domain representation. Stated differently: Every signal (video, audio, picture, etc.) has a frequency domain representation that is unique and contains all the information of the original signal. The way one computes the frequency domain representation from a given signal is called the Fourier transform Fourier transform definition (aside) 29

30 The spectrum We won t worry about analytically computing Fourier transforms in this class -- we ll save that treat for ECE Fundamentals III Instead, we will focus on some important properties and let software like MATLAB plot spectra for us. Analytical equations are generally unavailable for real-world signals, anyway Important properties: Linearity Symmetry 30

31 Linearity This property allows us to predict the spectrum of a combination of signals (e.g., a chord) from the spectra of the individual signals (e.g., pure tones). It can be divided into two elementary properties: Superposition: If x(t) has spectrum X(f), and y(t) has spectrum Y(f), then the sum x(t) + y(t) has the spectrum X(f) + Y(f). Scaling: If x(t) has spectrum X(f), and a is a constant, then, ax(t) has spectrum ax(f). Putting these together proves linearity: If x(t) has spectrum X(f), y(t) has spectrum Y(f), and a and b are both fixed, then ax(t) + by(t) has spectrum ax(f) + by(f). We ll see why linearity is an especially useful property in the next lecture. 31

32 Symmetry In general, given a complex-valued signal x(t), the spectrum X(f) has no guaranteed symmetry, and we need to look at both the positive and negative frequency axes to completely describe the signal. However, for a real-valued signal, the spectrum X(f) will have a special complex-valued symmetry called conjugate, or Hermitian, symmetry: The real component of the spectrum will have even symmetry: The imaginary component of the spectrum will have odd symmetry: The magnitude of the spectrum will have even symmetry: The phase of the spectrum will have odd symmetry: 32

33 Symmetry Therefore, if the signal is real-valued, the content on the positive and negative frequency axes are directly related to each other. Where we have a spike at +f, we will have a spike at -f. When MATLAB plots spectra of our signals, we will usually visualize just the positive frequency axis. 33

34 Some Fourier transform examples We already know a couple Fourier transforms: Here are a couple other useful ones: Decaying exponential Box function 34

35 Real world spectrum Here is an example of a real world spectrum (Hallelujah chorus): What do you notice about the predominant frequencies? Are all these frequencies being played at once? 35

36 Frequency vs. time: the spectrogram As we observed earlier, different sounds produce different frequency content. Music blends and varies different arrangements of these sounds through time. The Fourier spectrum provides a global picture of frequency content, it reflects all the frequencies present in the music. A more useful characterization of music would provide a local picture of frequency content as it varies through time. The spectrogram is one such characterization. Frequency (Hz) Mary had a little lamb, by Thomas Edison Time (s) 36

37 The spectrogram The spectrogram visualizes what we call the short-time Fourier transform. Normally, a single Fourier transform is computed for an entire signal. The short-time Fourier transform divides the signal into small, overlapping time intervals and computes a spectrum for each one separately. These spectra are plotted on a two-dimensional graph, with the frequency content along one axis, and the time dimension along the other. This 2D plot allows us to measure how the relative weighting of different frequency content changes through time. The spectrogram is one form of what is more generally called time-frequency analysis. Such methods allow engineers to design systems that analyze complicated signals in useful ways. Tools like Shazam form musical fingerprints from spectrograms to help identify music. Medical devices use time-frequency analysis to analyze heart rhythm. 37

38 Next time We will learn more about using spectra to study and process signals. Homework #2 and Lab #1 are due on Tuesday before class. 38

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