Sampling and Reconstruction6.003: Signal Processing Sampling and Reconstruction September 13, 2018...

6.003: Signal Processing

Sampling and Reconstruction

September 13, 2018

Importance of Discrete Representations

Our goal is to develop signal processing tools that help us under-

stand and manipulate the world.

Model Result

World New Understanding

make model

analyze

(math, computation)

interpret results

System Behavior

analysis

design

The increasing power and decreasing cost of computation makes the

use of computation increasingly attractive. However, many impor-

tant signals are naturally described with continuous domain.

→ understand the relation between CT and sampled signals

Sampling

Converting continuous-time (CT) signals to discrete time (DT).

t

x(t)

0T 2T 4T 6T 8T 10Tn

x[n] = x(nT )

0 2 4 6 8 10

T = sampling interval

How does sampling affect the information contained in a signal?

Sampling

We would like to sample in a way that preserves information.

However, information is clearly lost in the sampling process.

t

x(t)

From this example, it’s clear that uniformly spaced samples tell us

nothing about the values of x(t) between the samples.

Sampling

Even worse, information can be distorted, becoming misinformation.

Samples of an original signal x(t) = cos 7π3 n

t

cos 7π3 n? cos π3n?

are just as easily interpreted as coming from cos π3n.

Effects of Undersampling Are Easily Heard

Sampling Music

fs = 1T

• fs = 44.1 kHz

• fs = 22 kHz

• fs = 11 kHz

• fs = 5.5 kHz

• fs = 2.8 kHz

J.S. Bach, Sonata No. 1 in G minor Mvmt. IV. Presto

Nathan Milstein, violin

Effects of Undersampling are Easily Seen

Sampling Images

original: 2048× 1536


Sampling Images

downsampled: 1024× 768


Sampling Images

downsampled: 512x384


Sampling Images


Effects of Undersampling

If we hope to use sampled signals to understand behaviors of con-

tinuous signals, we must understand effects of sampling.

Effects of Undersampling

If we hope to use sampled signals to understand behaviors of con-

tinuous signals, we must understand effects of sampling.

Some of the effects of undersampling arise from fundamental prop-

erties of discrete time.

CT: x(t) = cosωt

DT: x[n] = cos Ωn

These signals differ in important ways.

Check Yourself

Compare two signals:

x1[n] = cos 3πn4

x2[n] = cos 5πn4

How many of the following statements are true?

x1[n] has period N=8.

x2[n] has period N=8.

x1[n] = x2[n].

Check Yourself

t

cos 3πt4 = cosω1t

t

cos 5πt4 = cosω2t

ω1 6= ω2

but x1[n] = x2[n]!

ω1 + ω2 = 2πcosω1n = cos(2π − ω2)n = cos(2πn− ω2n) = cos(−ω2n) = cosω2n

Check Yourself

Compare two signals:

x1[n] = cos 3πn4

x2[n] = cos 5πn4

How many of the following statements are true? 3

x1[n] has period N=8.√

x2[n] has period N=8.√

x1[n] = x2[n].√

Check Yourself

Consider 3 CT signals:

x1(t) = cos(4000t) ; x2(t) = cos(5000t) ; x3(t) = cos(6000t)

Each of these is sampled so that

x1[n] = x1(nT ) ; x2[n] = x2(nT ) ; x3[n] = x3(nT )

where T = 0.001.

Which list goes from lowest to highest DT frequency?

0. x1[n] x2[n] x3[n] 1. x1[n] x3[n] x2[n]

2. x2[n] x1[n] x3[n] 3. x2[n] x3[n] x1[n]

4. x3[n] x1[n] x2[n] 5. x3[n] x2[n] x1[n]

Check Yourself

The CT signals are simple sinusoids:


The DT signals are sampled versions (T = 0.001):

x1[n] = cos(4n) ; x2[n] = cos(5n) ; x3[n] = cos(6n)

How does the shape of cos(Ωn) depend on Ω?

Check Yourself

As the frequency Ω increases, the shapes of the sampled signals

deviate from those of the underlying CT signals.

n

Ω = 1 : x[n] = cos(n)

n

Ω = 2 : x[n] = cos(2n)

n

Ω = 3 : x[n] = cos(3n)

Check Yourself

Worse and worse representation.ΩL

n

Ω = 4 : x[n] = cos(4n) = cos(2π − 4n) ≈ cos(2.283n)

n


n


The same DT sequence represents many different values of Ω.

Check Yourself

For Ω > π, a lower frequency ΩL has the same sample values as Ω.

n


n

n


n

n


n

The same DT sequence represents many different values of Ω.

Check Yourself

Although there are multiple frequencies Ω with the same samples,

only one of these frequencies is in the range 0 ≤ Ω ≤ π.

Thus we can remove the ambiguity of which frequency is represented

by a set of samples by choosing the one in the range 0 ≤ Ω ≤ π.

We call that range of frequencies the base band of frequencies.

If we limit our attention to frequencies in the base band, then there

is a maximum possible discrete frequency Ωmax = π.

Note this difference from CT, where there is no maximum frequency.

Check Yourself

Consider 3 CT signals:


Each of these is sampled so that

x1[n] = x1(nT ) ; x2[n] = x2(nT ) ; x3[n] = x3(nT )

where T = 0.001.

Which list goes from lowest to highest DT frequency? 5

0. x1[n] x2[n] x3[n] 1. x1[n] x3[n] x2[n]

2. x2[n] x1[n] x3[n] 3. x2[n] x3[n] x1[n]

4. x3[n] x1[n] x2[n] 5. x3[n] x2[n] x1[n]

Relation Between CT and DT Frequencies

If a CT signal x(t) = cosωt is sampled at times t = nT , the resulting

DT signal is x[n] = cos Ωn where

Ω = ωT .

If we restrict DT frequencies to the range 0 ≤ Ω ≤ π, then the

corresponding CT frequencies are in the range 0 ≤ ω ≤ ωN where

ωN = π

T

and

fN = ωN2π = 1

2T = 12fs

where fN is the Nyquist frequency, which is half the sample rate fs.

Base Band Reconstruction

If all frequencies in a CT signal are less than the Nyquist frequency,

then we can reconstruct the original CT signal perfectly by discarding

frequencies in the reconstruction that exceed the Nyquist frequency.

This is called base-band reconstruction.

Anti-Aliasing

If there are frequencies in the CT signal that are greater than the

Nyquist frequency, then these will alias to frequencies in the base

band that have nothing to do with the original frequency.

These should be removed before sampling.

Anti-Aliasing Demonstration

Sampling Music

ωs = 2πT

= 2πfs

• fs = 11 kHz without anti-aliasing

• fs = 11 kHz with anti-aliasing

• fs = 5.5 kHz without anti-aliasing

• fs = 5.5 kHz with anti-aliasing

• fs = 2.8 kHz without anti-aliasing

• fs = 2.8 kHz with anti-aliasing



Quantization

The information content of a signal scales not only with sample rate

but also with the number of bits used to represent each sample.

11

0

0

1

1Input voltage

Outputvoltage 2 bits 3 bits 4 bits

00

01

10

0 0.5 11

0

1

Time (second)0 0.5 1

Time (second)0 0.5 1

Time (second)

1 0 1Input voltage

1 0 1Input voltage

Bit rate = (# bits/sample)×(# samples/sec)

Check Yourself

We hear sounds that range in amplitude from 1,000,000 to 1.

How many bits are needed to represent this range?

1. 5 bits

2. 10 bits

3. 20 bits

4. 30 bits

5. 40 bits

Check Yourself

How many bits are needed to represent 1,000,000:1?

bits range

1 22 43 84 165 326 647 1288 2569 512

10 1, 02411 2, 04812 4, 09613 8, 19214 16, 38415 32, 76816 65, 53617 131, 07218 262, 14419 524, 28820 1, 048, 576

Check Yourself

We hear sounds that range in amplitude from 1,000,000 to 1.

How many bits are needed to represent this range? 3

1. 5 bits

2. 10 bits

3. 20 bits

4. 30 bits

5. 40 bits

Quantization Demonstration

Quantizing Music

• 16 bits/sample

• 8 bits/sample

• 6 bits/sample

• 4 bits/sample

• 3 bits/sample

• 2 bit/sample



Quantizing Images

Converting an image from a continuous representation to a discrete

representation involves the same sort of issues.

This image has 280× 280 pixels, with brightness quantized to 8 bits.

Quantizing Images

8 bit image 7 bit image

Quantizing Images


Check Yourself

What is the most objectionable artifact of coarse quantization?


Dithering

One very annoying artifact is banding caused by clustering of pixels

that quantize to the same level.

Banding can be reduced by dithering.

Dithering: adding a small amount (±12 quantum) of random noise to

the image before quantizing.

Since the noise is different for each pixel in the band, the noise

causes some of the pixels to quantize to a higher value and some to

a lower. But the average value of the brightness is preserved.

Quantizing Images with Dither

7 bit image 7 bits with dither


1 bit image 1 bit with dither

Check Yourself

What is the most objectionable artifact of dithering?

3 bit image 3 bit dithered image

Check Yourself


One annoying feature of dithering is that it adds noise.

Quantization Schemes

Example: slowly changing backgrounds.

Quantization: y = Q(x)

Quantization with dither: y = Q(x+ n)

Check Yourself


One annoying feature of dithering is that it adds noise.

Robert’s technique: add a small amount (±12 quantum) of random

noise before quantizing, then subtract that same amount of random

noise.

Quantization Schemes

Example: slowly changing backgrounds.

Quantization: y = Q(x)

Quantization with dither: y = Q(x+ n)

Quantization with Robert’s technique: y = Q(x+ n)− n

Quantizing Images with Robert’s Method

7 bits with dither 7 bits with Robert’s method


1 bits with dither 1 bit with Robert’s method

Quantizing Images: 3 bits

8 bits 3 bits

dither Robert’s

Quantizing Images: 2 bits

8 bits 2 bits

dither Robert’s

Quantizing Images: 1 bit

8 bits 1 bit

dither Robert’s

Sampling and Reconstruction6.003: Signal Processing Sampling and Reconstruction September 13, 2018...

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