Module 2: GPS - University of Virginiaffh8x/d/soi18F/Module02.pdfSampling theorem Perfect...

2.1

MODULE 2: Frequency domain representation and sampling in

the iPhone

Motivation

Signal spectrum

Sampling

Undersampling, oversampling, critical sampling and aliasing

Sampling theorem

Perfect reconstruction

credit: xkcd

2.2

Information in the real world is analog. To store them on an iPhone, we sample the signal:

When stored, we only have the samples:

2.3

Can we recover the original signal? There are many analog signals (an infinite number,

actually) that fit these samples:

2.4

The Frequency Domain

Motivation

The frequency domain is the key to:

o Determining how many samples the iPhone needs to take of a sound or an image or a

video.

o Being able to perfectly represent and reconstruct a signal (voice, music, image,

video)

o Being able to compress signals.

Ex. 64 GB iPhone; let’s say 10 GB are for music.

Without compression (CD quality), you could fit 185 songs on your iPhone.

That’s maybe your collection of songs by _________.

With compression (mp3), you could fit about 2500 songs. About 166 hours of

playing time to shuffle the full iPhone.

o Being able to communicate with the iPhone. That’s MODULE 5…

2.5

We generally think of a signal as a function of time f(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.

Basic idea: A signal can be represented by a set of sinusoids (e.g., a set of cosine

functions).

A sinusoid of a given frequency can be taller or shorter (magnitude) or shifted left or right

(phase).

The description of a signal by frequency content is quite indispensable.

Sinusoids

Model: 𝐴 𝑐𝑜𝑠 (2𝜋𝑓𝑡 + 𝜙)

𝐴: amplitude

2.6

𝑓: frequency in Hertz – cycles per second = 1/𝑇 = 1/Period

2𝜋𝑓: radial frequency in radians per second

𝜙: phase in radians (offset from origin)

cos (𝜔𝑡 −𝜋

2) = sin(𝜔𝑡)

Ex: 1.0 cos(2𝜋440𝑡) frequency = 440Hz = ‘A’

Listen to it (*pureA.wav)

2.7

We’re actually used to thinking of some signals this way

o Music: A = 440 Hertz (what does this mean?)

o Light: EM Waves at different frequencies – ROYGBIV. White = sum

We can model these waves using sinusoids.

o Graphic Equalizer: selectively boost or attenuate sounds within a certain frequency

range (e.g., iTunes)

o Radio tuning: select a station at a particular frequency

2.8

Frequency domain is important from both physical and mathematical points of view:

The physical world reacts to different frequencies differently.

Our senses respond differently to different frequencies:

Eyes: different cones responding to different colors

2.9

Ears: “Hair” cells respond to different frequencies:

This is not limited to our senses but all Linear Time-Invariant (LTI) systems (more on

this later).

Mathematically the frequency domain allows us to formulate certain problems in a much

simpler way, for example sampling and perfect reconstruction.

2.10

Back to the spectrum

Plot of amplitude (magnitude) vs. frequency

o The spectrum of 1.0cos(2440t) is simpler than it’s amplitude

(Note: in this class, we will deal only with the magnitude spectrum. There is also a ‘phase

spectrum’, just as every sinusoid has both a magnitude and a phase. More in Fun III

(ECE)…)

2.11

Time Domain: x(t) = A cos(2π440t)

Frequency Domain: X(f) = A δ(f – 440)

o δ(f) is a spike – more formally, a delta function

o Job security term: Dirac delta function

Aside: What are the properties of this delta function?

– δ(t) = 0 for t ≠ 0

– δ(0)is undefined but δ(t) has unit area

∫ 𝛿(𝑡)𝑑𝑡 = 1∞

−∞

Bottom line:

o Spectrum for cos(2𝜋𝑓0𝑡) consists of a spike at 𝑓0

o Spectrum for sin(2𝜋𝑓0𝑡) is similar; it also consists of a spike at 𝑓0

2.12

More about the delta function (Optional)

𝛿(𝑡) = lim𝜖→0

𝑃𝜖(𝑡)

P(t)

t

휀 = 1 휀 = 1/2 휀 = 1/4

2.13

What happens if multiply the delta function by another function? Let’s see that with 𝑃𝜀.

What is the area under the curve, if 휀 is very small?

∫ 𝑔(𝑡)𝛿(𝑡)𝑑𝑡 = 𝑔(0)∞

−∞

∫ 𝑔(𝑡)𝛿(𝑡 − 𝑡0)𝑑𝑡 = 𝑔(𝑡0)∞

−∞

The spectrum for cos(2𝜋𝑓0𝑡), if it extends from −∞ to +∞, is 𝛿(𝑓 − 𝑓0)

o In the examples that we will see, the time span is limited to the spectrum is an

approximation of the delta function

𝑔(𝑡)

𝑔(0)

𝑃𝜀(𝑡) 𝑃𝜀(𝑡)𝑔(𝑡)

𝑔(0)/(2휀)

1/(2휀)

2.14

Let’s look at a slightly more complicated signal than the last example… two tones that your

iPhone can play simultaneously…

1

2(cos(2𝜋440𝑡) + cos (2𝜋554.4𝑡))

(Listen: *pureAthird.wav)

2.15

1

3(cos(2𝜋440𝑡) + cos(2𝜋554.4𝑡) + cos (2𝜋659.3𝑡))

Listen: *pureAtriad.wav; three tones for your iPhone

2.16

1

4(cos(2𝜋440𝑡) + cos(2𝜋554.4𝑡) + cos(2𝜋659.3𝑡) + cos(2𝜋880𝑡))

Listen: *pureAchord.wav

2.17

21/12 = 1.0595

Virtuoso app on iPhone:

440 Hz 880 Hz

A B C D E F G A B C D E F G A

1760 Hz

one octavefrequency doubles

one semitonefrequency increases by 2 1/12

each octave has12 semitones

2.18

Listen to the A chord: *pureAchord.wav (synthetic)

*pianoA.wav (keyboard)

440 Hz

A C# E A

440 x 24/12 = 554.4 Hz

12 semitones = 1 octave

7 semitones = “perfect fifth”

4 semitones = “major third”

440 x 212/12 = 880 Hz

440 x 27/12 =659.3 Hz

2.19

Sounds better. What’s the difference with the pure A (computerized)?

Piano A

2.20

Musical instruments (such as the piano) do not generate pure sinusoids, but contain

harmonics.

2.21

Harmonics

Take A, for example:

o A = 440 Hz = fundamental frequency

o 880 Hz = 2f = Second Harmonic (note typical faux pas)

o 1320 Hz = 3f = Third Harmonic

o etc.

Job security (also way to embarrass a colleague): first “overtone” is the second harmonic.

Bottom line: the spectral description (the spectrum) is much simpler than the time domain

representation.

The spectrum is also more informative, more descriptive in this case.

Ex: *trumpetA.wav… How would you describe this sound? (compared to piano, for

example)… Let’s look at the spectrum

2.22

What’s odd about this spectrum?

Trumpet A

2.23

Same fundamental frequency (look at basic period of repetition), but different appearance in

time domain… how would you describe it?

Piano

Trumpet

2.24

Let’s consider two notes (A, C#) = major third (*pianoAthird.wav)

Piano A + C#

2.25

*trumpetAthird.wav (same frequencies, different amplitudes)

Trumpet A + C#

2.26

One observation from the previous examples: adding an A and a C# did not produce

any “new” frequencies, just A, C# and harmonics. This is a consequence of the linearity

– that we can analyze the frequency components separately – the spectrum is the sum of

the spectra from each component.

Mathematically,

o if 𝑋(𝑓) is the spectrum of 𝑥(𝑡)

o and 𝑌(𝑓) is the spectrum of 𝑦(𝑡),

o then 𝑋(𝑓) + 𝑌(𝑓) is the spectrum of 𝑥(𝑡) + 𝑦(𝑡).

Let’s look at some more complicated spectra. *mandolin

2.27

Spectrum is more complicated. Why?

We can say in general that

Mandolin

2.28

o Purely periodic signals have discrete spectra (consisting of delta functions at different

frequencies)

o Non-periodic signals have continuous spectra

2.29

Quantization errors and frequency domain

Listen to guitar_bass and guitar_bass2

2.30

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…

(Note: I call these tenets “Science of Information” tenets, as they extend far beyond the

iPhone.)

Science of Information Fundamental Tenet I:

There is a one-to-one relationship between a signal and its

frequency domain representation.

LOSSY IMAGE CODING

• We will review a few of popular (among many) approaches.

- (1) Vector Quantization Coding

- (2) Discrete Cosine Transform (DCT) Coding and JPEG

- (3) Wavelet-Based Image Coding

The compression ratios obtained over the first 20 years of

research:

• Things have changed!

Block Coding of Images

• Most image coding methods, including the JPEG standard,

involve breaking the image into sub-blocks to be

2.31

Brief Discussion of Fourier Representation

Consider 𝑥(𝑡) = 𝐴1 cos(2𝜋𝑓1𝑡) + 𝐴2 cos(2𝜋𝑓2𝑡) + 𝐴3 cos(2𝜋𝑓3𝑡) + ⋯

Periodic signals can be written as sum of sinusoids.

We call this a Fourier series representation. The A’s are the Fourier coefficients.

This matches well to the discrete spectrum we discussed.

Periodic and non-periodic signals can be written as an integral of sinusoids

𝑥(𝑡) = ∫ 𝑋(𝑓) cos(2𝜋𝑓𝑡)𝑑𝑓∞

0

(More precisely 𝑥(𝑡) = ∫ |𝑋(𝑓)| cos(2𝜋𝑓𝑡 + ∠𝑋(𝑓))𝑑𝑓∞

0)

This is a Fourier integral representation.

This works well for the continuous spectrum we discussed.

2.32

I don’t want to spoil your future classes on this subject (!), so we’ll avoid the math for right

now.

Just one example:

o For 𝑥(𝑡) = cos(2𝜋𝑓0𝑡), 𝑋(𝑓) = 𝛿(𝑓 − 𝑓0)

o 𝑥(𝑡) = ∫ 𝛿(𝑓 − 𝑓0)+∞

0cos(2𝜋𝑓𝑡) = cos(2𝜋𝑓0𝑡).

Bottom line: there is a direct relationship between 𝑥(𝑡) and 𝑋(𝑓), the signal and its

spectrum… this is the Fourier Transform.

Spectral analysis is very useful.

o EX: we can’t hear above 20KHz. So stereo doesn’t need to reproduce sound over this

limit.

o Speech is limited to 3-4 KHz. So the telephone doesn’t carry components above 4

KHz.

The iPhone needs the spectral representation to communicate (more on this later).

The iPhone needs the spectral representation to compress and store signals.

2.33

Recall the Mandolin spectrum, most of the components were < 1KHz. Listen to

*soprano.wav.

Note: The above graph (in purple) shows the magnitude spectrum, which tells how much

of each sinusoid is in the signal. There is also the phase spectrum, which tells how much

Soprano

2.34

each sinusoid is shifted in the signal (effectively, where the sinusoid starts). We will

concentrate on magnitude, but phase is a critical component of the frequency domain.

2.35

Question: what would happen if we doubled the speed of playback?

*sopranodoubletime.wav

The peak at 900 Hz is now at 1800Hz

If 𝑋(𝑓) is the spectrum of 𝑥(𝑡), then the spectrum of 𝑥(2𝑡) is 𝑋(𝑓/2).

Soprano - speed doubled

2.36

Conclusion: changing the speed of a musical recording changes pitch. To change speed

without altering pitch is not straightforward!

*voice spinner in chrome music lab

2.37

Listen to this baritone (*terfel1.wav)

How does the spectrum compare to that of the soprano?

Baritone

2.38

Listen to *carminaburana.wav

Combination of musical instruments and voice produces rich spectrum.

Carmina Burana

2.39

Let’s listen to *berlioz.wav

Although complicated, there are distinct peaks.

Can we determine the key from the spectrum?

Berlioz

2.40

Major triad: B♭, D, F (B♭is 440 x 21/12), Other peaks: notes in Bb major scale (C,

E♭, G, A)

Bb D F

Bb D F

2.41

* spectrogram in chrome music lab

How does Shazam or SoundHound work on your iPhone?

Shazam (for example) makes a spectral fingerprint of each song.

It compares your recording to a set of fingerprints and then attempts to match.

The fingerprints are derived from the spectrum.

But this spectrum is changing in time; so Shazam uses the spectrogram – a graph of

spectral plots through time.

2.42

Consider two sinusoids: x1(t) = cos(2π2093t) = C, x2(t) = cos(2π3520t) = A

octave up

Listen to them (sampled at 44100 hertz) *sig3a_44100.wav, sig3b_44100.wav

2.43

𝑥𝑠𝑢𝑚(𝑡) = x1(t) + x2(t) = cos(2π2093t) + cos(2π3520t)

The spectrum is just the sum of the spectra. Superposition.

Listen to *sig3c_44100.wav

2.44

𝑥𝑝𝑟𝑜𝑑𝑢𝑐𝑡(𝑡) = x1(t)x2(t) = cos(2π2093t) cos(2π3520t)

Multiplication: listen to *sig3d_44100.wav – not original tones!

2.45

Look at spectrum – two components, but not at original frequencies (2093, 3520)

2.46

Explanation

Trig identity: cos(𝑎) cos(𝑏) =1

2(cos(𝑎 + 𝑏) + cos(𝑎 − 𝑏))

So, multiplication of two cosines produces sum and difference frequencies.

In the example we just looked at 𝑓1 + 𝑓2 = 2093 + 3520 = 5613 𝐻𝑧, 𝑓1 − 𝑓2 = 1427 𝐻𝑧

𝑥𝑝𝑟𝑜𝑑𝑢𝑐𝑡(𝑡) = 𝑥1(𝑡)𝑥2(𝑡) = cos(2π2093t) cos(2π3520t)

=1

2(cos(2𝜋(3520 − 2093)𝑡) + cos(2𝜋(3520 + 2093)𝑡))

=1

2(cos(2𝜋1427𝑡) + cos(2𝜋5613𝑡))

Is this what we’re hearing? Let’s check

o *sig3e_44100 is the 1427 Hz tone

o *sig3f_44100 is the 5613 Hz tone

2.47

Spectra of 1427 and 5613 Hz tones:

2.48

How does this sum and the product compare?

Hmmmm…. They look pretty similar

2.49

Negative frequencies

In the last example, we could have subtracted 3520 from 2093 to get a frequency of -1427

Hz … and this would have been OK

Remember cos(𝑎) = cos (−𝑎) – it’s an even function

o Therefore, cos(2𝜋𝑓𝑡) = cos(−2𝜋𝑓𝑡)

The spectrum has a negative side, which is the mirror image of the positive side, so we

normally do not show it.

Multiplying by the cosine, again

2.50

What happens if we multiply the signal by cos(2𝜋𝑓1𝑡) ?

Remember if we multiplied another cosine with frequency 𝑓0 by the above cosine,

we would see frequencies of 𝑓0 − 𝑓1 and 𝑓0 + 𝑓1.

In general, we get two copies, one shifted by 𝑓1 to the right, and one shifted by 𝑓1 to

the left

Full spectrum of signal multiplied by cos(2𝜋𝑓1𝑡)

𝑓0 −𝑓0

Positive spectrum Full spectrum

2.51

Linear Time Invariant Systems and Filtering

A system is any process that given an input signal produces an output signal (almost

anything):

A microphone: input: sound waves, output: voltage

A speaker: input: voltage, output: sound waves

A car: input: pressure on gas pedal, output: position

A linear time-invariant system:

1. responds to a signals linearly:

𝑥(𝑡) → 𝑦(𝑡), 2𝑥(𝑡) → 2𝑦(𝑡)

EX: Car goes twice as fast if you push the gas pedal twice as much

2. is time-invariant: if we shift the input in time, the output is also shifted by the same

amount

EX: tapping on the desk creates the same sound today and tomorrow.

The iPhone’s mic and speaker are LTI systems.

2.52

The response of LTI systems to a sinusoid is multiplication by a factor that depends on

the frequency of the sinusoid:

𝑥(𝑡) = 𝐴 cos(2𝜋𝑓𝑡) → 𝑦(𝑡) = 𝐻(𝑓)𝐴 cos(2𝜋𝑓𝑡 + 𝜙)

𝐻(𝑓) > 1: amplification

𝐻(𝑓) < 1: attenuation

EX:

Response of an LTI system to a sinusoid; signal is amplified

Input Output

2.53

EX: Response of an LTI system to a sinusoid; signal is attenuated

Input Output

2.54

Response of an LTI system to a non-sinusoid; the shape of the signal changes

2.55

When the input is a sinusoid, it may be amplified or attenuated, depending on its frequency.

This is determined by the spectral response of the system.

Source: FaberAcoustical

Spectral response of Microphone from Three iPhones

2.56

Filtering

When the signal spectrum consists of different frequencies, each frequency is treated

differently by the system (may be attenuated or amplified).

A filter is a system designed to manipulate the frequency components of a signal.

Low pass filter: A filter that allows

low frequencies to go through but

eliminates high frequencies

High pass filter: A filter that allows

high frequencies to go through but

eliminates low frequencies

2.57

Consider:

𝑥(𝑡) =1

3(cos(2𝜋250𝑡) + cos(2𝜋500𝑡) + cos (2𝜋1000𝑡))

* Listen to FLT_threetones.wav

2.58

Low pass filter

2.59

Low pass filter

2.60

High pass filter

2.61

5 Hz Sawtooth Signal

2 5t floor 5t( )– y(t)

Y(f)

FFT

2.62

Reconstruction of time-domain signal from frequency spectrum

Y(f)

y(t)

IFFT

2.63

Band limiting a signal (passing through a low pass filter):

2

n------ 2n5t

2---+

cos

n 1=

10

1+

n = 1 fundamental frequency

n = 10 10th harmonic

n = 0 dc term

2.64

Reconstruction of time-domain signal from frequency spectrum

Y(f)

y(t)

IFFT

2.65

Toward the Sampling Theorem

How does one get a digital signal inside the iPhone from an audio signal (e.g., from the

microphone)?

Answer: Analog-to-Digital

conversion

In the iPhone, this is integrated in

the microphone:

Source: iFixit

Knowles S1950 microphone in iPhone 4

2.66

The iPhone uses a mixed signal (analog and digital) ASIC (application specific integrated

circuit) to do A-to-D conversion… inside the microphone!

Recall: Sampling is the process of taking values of the signal at discrete points.

The above microphone in the iPhone uses discrete-time sampling.

The digital camera in the iPhone utilizes discrete-space sampling.

Another question: how often does the iPhone camera unit need to sample?

o To make a perfect digital replica.

o To minimize the number of samples.

Source: ChipWorks

Knowles S1950 ASIC in iPhone 4

2.67

A picture is sampled in pixels/cm or dots per inch (dpi)…

Higher sampling rate, better fidelity, but more pixels to store and transmit and process…

Audio Question:

Suppose we have a pure tone signal: x(t) = cos(2πft)

128 x 94 pixels 64 x 47 pixels

32 x 24 pixels 16 x 12 pixels

2.68

The period of the cosine is 𝑇 =1

𝑓 – we often think of this period being broken up into 360

degrees or 2 radians.

For reasonable reconstruction, how many samples per period should we take?

What happens if we sample above or below this rate?

Listen to *sig28000.wav (sum of two sinusoids and 2KHz and 3.364KHz – major 6th)

Compare to *sig244100.wav – same signal, sampled much faster (>5X)

o This signal takes > 5X the storage… is it worth it?

o Moral: oversampling does not improve fidelity

Undersampling: let’s listen to *sig24000.wav

o Do you hear any new tones introduced?

o Why did this happen?

o Need to look at what is happening in the frequency domain.

2.69

Generation of a Sampled Signal

2.70

Signal sampled at 2048 Hz

Sampling at frequency 𝑓𝑠 can be viewed as multiplying by a sequence of spikes that are 1/𝑓𝑠

apart

2.71


Let’s increase the sampling rate to 4096 Hz, 8192 Hz, 16384 Hz

2.72


2.73


2.74


2.75

Sampling Function

fs 27

samples/s=

2.76

Example: Sampling a band-limited saw-tooth signal:

Sampling Process

×

=

2.77

We know what happens in the time domain, but what about the frequency domain?

Sampled and Original Bandlimited Signal

fs 27

samples/s=

2.78

A repeating pattern! Let’s compare with the spectrum of the non-sampled signal.

Sampled Signal and Spectrum

fs 27

samples/s=

2.79

The spectrum is consists of shifts of the original spectrum. Where have we seen shifts of

the spectrum before?

Sampled Signal Contains Spectral Replicas

fs 27

samples/s=

128 Hz

256 Hz384 Hz

2.80

But first, what is the effect of the sampling rate?

Spectral Replicas Occur at Multiples of fs

fs 27

samples/s=

128 Hz

2.81

Effect of Increasing Sampling Frequency fs

fs 27

samples/s= fs 28

samples/s=

2.82

Effect of Decreasing Sampling Frequency fs

fs 27

samples/s= fs 26

samples/s=

2.83

Multiplying by a cosine creates a shift in the frequency domain! But what does the

sampling function have to do with cosines? They don’t look similar at all!

Cosine at fs

Cosine at 𝑓𝑠 Sampling function at 𝑓𝑠

2.84

Cosines at fs , 2fs

2.85

Cosines at fs , 2fs , 3fs

2.86

Sum of Cosines at fs , 2fs , 3fs

2.87

Normalized Sum of Cosines at f = 0, fs , 2fs ... Mfs

1

M----- 2nfst cos

n 0=

M

M 3=

2.88


1

M----- 2nfst cos

n 0=

M

M 5=

2.89


1

M----- 2nfst cos

n 0=

M

M 15=

2.90


1

M----- 2nfst cos

n 0=

M

M 666=

2.91

The sampling function is actually the sum of many cosines, at frequencies that are

multiples of 𝑓𝑠

Sampling Function and Spectrum

2.92

So multiplying by the sampling function is the same as multiplying by those cosines

Sampling Process in the Time Domain

×

=

2.93

In the frequency domain, the spectrum of the signal is shifted, once for each of the

cosines. (the technical term for what happens in the frequency domain is convolution)

Sampling Process in the Frequency Domain

*

=

2.94

cos(2𝜋𝑓𝑡) × cos(2𝜋𝑓𝑠𝑡) =1

2(cos(2𝜋(𝑓 − 𝑓𝑠)𝑡) + cos(2𝜋(𝑓 + 𝑓𝑠)𝑡))

→ Multiplying by cosine shifts the spectrum by ±𝑓𝑠 (to the right and left by 𝑓𝑠)

Likewise,

Signal × cos(2𝜋𝑓𝑠𝑡) shifts all the frequencies in the signal by ±𝑓𝑠, creating

two copies of the spectrum one shifted by 𝑓𝑠 to the right and one to the left

Therefore,

Signal × sampling function shifts all the frequencies in the signal spectrum by

0, ±𝑓𝑠, ±2𝑓𝑠, ±3𝑓𝑠, …

2.95

Reconstruction from spectra of sampled signals

27 Hz sampling rate 28 Hz sampling rate

2.96

×

=

2.97

Reconstruction from filtered spectrum of signal sampled at 27 Hz

Perfect reconstruction of a continuous signal from a finite number of discrete samples.

2.98

Effect of Increasing Sampling Frequency fs

fs 27

samples/s= fs 28

samples/s=

2.99

×

=

2.100

Effect of Decreasing Sampling Frequency fs

fs 27

samples/s= fs 26

samples/s=

2.101


Oversampling does not improve fidelity.

2.102

×

=

2.103


Imperfect reconstruction results when the sampling rate is insufficiently high.

2.104

So, the result in the frequency domain is (an infinite number of) copies of our original

spectrum repeated every fs Hz.

0 W - W

f

Original signal with bandlimit = W

0 W - W

fs -fs f

Sampled signal, sampled at fs

2.105

Basic principle: If you have a signal with a spectrum centered at zero frequency (=d.c., for

job security reasons), then the periodic replicas in the spectrum of the sampled signal occur

at multiples of fs.

What happens if we sample more frequently (shorter sampling period, larger sampling

frequency)?

o The replicas of the original spectrum become farther apart.

Science of Information Fundamental Tenet II:

Sampling a continuous signal at frequency fs results in a new

spectrum that contains an infinite number of copies of the

original spectrum separated by intervals of fs.

2.106

Amazing Result: Perfect Reconstruction

Example: iPhone 7

We can represent a signal digitally with perfect fidelity!

With a finite number of samples, we can perfectly represent the analog signal!

Source: https://tinhte.vn/

The inclusion of a digital-to-analog converter, or DAC for short,

enables both the new EarPods and traditional analog headphones with

3.5mm jacks to function over the Lightning connector, which delivers digital audio.

Lightning EarPods and Lightning-to-3.5mm

headphone jack adapter reveals D-to-A converter

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Application: D to A conversion… making an analog signal out of a digital one

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Requirements for Perfect Reconstruction:

o Signal must be bandlimited (that's OK – it's natural)

o Replicas cannot overlap with original spectrum

If the frequency range of a signal is from 0 to W, the bandwidth is W.

o Example W for audio = 20 KHz

o If not bandlimited, the replicas would overlap (with each other and with original

spectrum).

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The sampling frequency (fs) and the bandwidth of the signal (W) determine the spacing

between replicas.

0 W - W

fs -fs f

Oversampled

0

fs -fs f

Undersampled (aliased)

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W -W 0 fs -fs f

Critically sampled

Rule: fs – W > W OR fs > 2W (this is the sampling theorem!)

We have to sample at a frequency that is greater than twice the bandwidth (greater

than twice the highest frequency in the signal).

2W is called the Nyquist frequency (for job security).

Science of Information Fundamental Tenet III (Sampling Theorem):

Perfect reconstruction of a bandlimited signal from a finite number of

samples is possible by sampling above the Nyquist frequency.

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AFLAC trivia question: Remember when I asked about sampling the cosine: cos (2𝜋𝑓𝑡) –

how many samples per period.

ANS: Only 2!

W=f; 2W=2f, need > 2f samples / s; there are f periods / s.

Bonus: I said 2, but the answer is actually "more than two". How could you, in

sampling a sinusoid with two samples per period, get "unlucky"???????????? (let's draw

it)

Extra bonus trivia question: in the digital world, what's the fastest signal (highest

frequency signal) you can make?

To be on the safe side, we usually sample above the Nyquist frequency.

o Because ideal LP filters aren't ideal in the real world (let's draw some).

o We call the space in between the replicas "guard bands," again for job security.

Ex: Standard audio upper limit = 20 KHz, but CD uses 44.1 KHz for sampling.

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Aliasing

When we undersample (too slow, at or below 2W samples / s), the spectra (the replicas)

overlap.

When they overlap, we try to foolishly recover the original spectrum

But our low pass filter grabs the original spectrum AND some stuff we don't want (the

overlap from adjoining replicas).

We cannot easily eliminate this false information.

It's called ALIASING.

(let's look at an example…)

1

2(cos(2𝜋5613𝑡) + cos(2𝜋1427𝑡))

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Listen to this signal sampled at 44.1KHz (*sig3g_44100.wav); what's Nyquist?

(is 16384 above Nyquist? *sig3g_16384.wav)

What about 8192 Hz? Listen: *sig3g_8192.wav

After sampling at 8192 Hz, spectral replicas appear here

8192 Hz

8192 Hz

1427 5613-1427-5613

-5613 + 8192 = 2579 Hz

-1427 + 8192 = 6765 Hz

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Remember: 𝑊 > 𝑓𝑠 − 𝑊; 𝑓𝑠 > 2𝑊

What tones are we getting? (1427 Hz and 2579 Hz: *sig3h_8192.wav)

𝟏

𝟐(𝐜𝐨𝐬(𝟐𝝅𝟓𝟔𝟏𝟑𝒕) + 𝐜𝐨𝐬(𝟐𝝅𝟏𝟒𝟐𝟕𝒕)) sampled at 8912 Hz

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This spectrum proves our point: after undersampling, we get two sinusoids (one is false!)

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Reconstruction from 1024 Hz sampling rate

Original Signal and Spectrum



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Now let's look at a real signal…

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*wienerblut8192.wav

Wiener Blut, sampled at 8192 Hz

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After LPF at 1024 Hz (*wienerblut8192filtered.wav)

Wiener Blut, sampled at 8192 Hz, lowpass filtered

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Look at region below 500 Hz... what's happened? (*wienerblut2048.wav)

Wiener Blut, sampled at 2048 Hz

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Summary

A bandlimited signal has highest frequency component at W Hz

Bandwidth is 0 to W = W

If signal is not bandlimited in the first place, we can make it bandlimited using a low pass

filter.

We have to sample the signal at 𝑓𝑠 > 2𝑊

2W is the Nyquist frequency.

Perfect Reconstruction is possible by sampling at sufficiently high rate.

Sampling below the Nyquist frequency results in aliasing. False information.

Aliasing also occurs in images and movies

*wheel12a.avi (12 samples /s ) etc. (temporal)

*spiral.avi (spatial)

Module 2: GPS - University of Virginiaffh8x/d/soi18F/Module02.pdfSampling theorem Perfect...

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