ia802807.us.archive.org · 2018. 8. 12. · the sampling theorem the space of Ω n-bandlimited...

Digital Signal Processing

Module 6: Interpolation and SamplingDigital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

Module Overview:

◮ Module 6.1: Continuous-time signals

◮ Module 6.2: Interpolation

◮ Module 6.3: Sampling

◮ Module 6.4: Aliasing

◮ Module 6.5: Interpolation and sampling in practice

◮ Module 6.6: Discrete-time processing of continuous-time signals

6 1

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

Module 6.1: The Continuous-Time ParadigmDigital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

Overview:

◮ Models of the world

◮ Continuous-time signals

6.1 2

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

Overview:

◮ Models of the world

◮ Continuous-time signals

6.1 2

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

Two views of the world

Analog/continuous versus discrete/digital

6.1 3

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

Digital processing of signals from/to the analog world

◮ input is continuous-time: x(t)

◮ output is continuous-time: y(t)

◮ processing is on sequences: x [n], y [n]

analog world processing analog world

examples: MP3, digital photography

6.1 4

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

Digital processing of signals to the analog world

◮ input is discrete-time: x [n]

◮ output is continuous-time: y(t)

digital worldprocessingfor synthesis

analog world

examples: computer graphics, video games

6.1 5

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

Digital processing of signals from the analog world

◮ input is continuous-time: x(t)

◮ output is discrete-time: y [n]

analog worldprocessingfor analysis

digital world

examples: control systems, monitoring

6.1 6

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

digital worldview:

◮ arithmetic

◮ combinatorics

◮ computer science

◮ DSP

analog worldview:

◮ calculus

◮ distributions

◮ system theory

◮ electronics

6.1 7

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

digital worldview:

◮ countable integer index n

◮ sequences x [n] ∈ ℓ2(Z)

◮ frequency ω ∈ [−π, π]

◮ DTFT: ℓ2(Z) 7→ L2([−π, π])

analog worldview:

◮ real-valued time t (sec)

◮ functions x(t) ∈ L2(R)

◮ frequency Ω ∈ R (rad/sec)

◮ FT: L2(R) 7→ L2(R)

6.1 8

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

Bridging the gap

x [n] sound card

Ts system clock

6.1 9

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

Bridging the gap

sound card x [n]

Ts system clock

6.1 10

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

Bridging the gap

x [n]

sampling

x(t)

interpolation

6.1 11

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

About continuous time

◮ time: real variable t

◮ signal x(t): complex functions of a real variable

◮ finite energy: x(t) ∈ L2(R)

◮ inner product in L2(R)

〈x(t), y(t)〉 =∫ ∞

−∞

x∗(t)y(t)dt

◮ energy: ||x(t)||2 = 〈x(t), x(t)〉

6.1 12

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

〈x(t), y(t)〉 =∫ ∞

−∞

x∗(t)y(t)dt

◮ energy: ||x(t)||2 = 〈x(t), x(t)〉

6.1 12

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

〈x(t), y(t)〉 =∫ ∞

−∞

x∗(t)y(t)dt

◮ energy: ||x(t)||2 = 〈x(t), x(t)〉

6.1 12

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

〈x(t), y(t)〉 =∫ ∞

−∞

x∗(t)y(t)dt

◮ energy: ||x(t)||2 = 〈x(t), x(t)〉

6.1 12

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

〈x(t), y(t)〉 =∫ ∞

−∞

x∗(t)y(t)dt

◮ energy: ||x(t)||2 = 〈x(t), x(t)〉

6.1 12

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

Analog LTI filters

x(t) H y(t)

y(t) = (x ∗ h)(t)= 〈h∗(t − τ), x(τ)〉

=

∫ ∞

−∞

x(τ)h(t − τ)dτ

6.1 13

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

Analog LTI filters

x(t) H y(t)

y(t) = (x ∗ h)(t)= 〈h∗(t − τ), x(τ)〉

=

∫ ∞

−∞

x(τ)h(t − τ)dτ

6.1 13

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

Analog LTI filters

x(t) H y(t)

y(t) = (x ∗ h)(t)= 〈h∗(t − τ), x(τ)〉

=

∫ ∞

−∞

x(τ)h(t − τ)dτ

6.1 13

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

Fourier analysis

◮ in discrete time max angular frequency is ±π

◮ in continuous time no max frequency: Ω ∈ R

◮ concept is the same:

X (jΩ) =

∫ ∞

−∞

x(t)e−jΩtdt ← not periodic!

x(t) =1

2π

∫ ∞

−∞

X (jΩ)e jΩtdΩ

6.1 14

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

Fourier analysis

X (jΩ) =

∫ ∞

−∞

x(t) =1

2π

∫ ∞

−∞

X (jΩ)e jΩtdΩ

6.1 14

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

Fourier analysis

X (jΩ) =

∫ ∞

−∞

x(t) =1

2π

∫ ∞

−∞

X (jΩ)e jΩtdΩ

6.1 14

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

Real-world frequency

◮ Ω expressed in rad/s

◮ F =Ω

2π, expressed in Hertz (1/s)

◮ period T =1

F=

2π

Ω

6.1 15

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

Example

x(t) = e−t2

2σ2

−50 −40 −30 −20 −10 0 10 20 30 40 500

1

x(t)

6.1 16

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

Example

X (jΩ) = σ√2πe−

σ2

2Ω2

σ√2π

−50 −40 −30 −20 −10 0 10 20 30 40 50

X(jΩ)

6.1 17

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

Convolution theorem

x(t) H y(t)

Y (jΩ) = X (jΩ)H(jΩ)

6.1 18

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

A new concept: bandlimited functions

ΩN -bandlimitedness:

X (jΩ) = 0 for |Ω| > ΩN

6.1 19

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

Prototypical bandlimited function

0−ΩN ΩN

G

6.1 20

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

The prototypical bandlimited function

Φ(jΩ) = G rect

(

Ω

2ΩN

)

ϕ(t) =1

2π

∫ ∞

−∞

Φ(jΩ)e jΩtdΩ

= . . . see Module 5.5

= GΩNπ

sinc

(

ΩNπ

t

)

6.1 21

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

Φ(jΩ) = G rect

(

Ω

2ΩN

)

ϕ(t) =1

2π

∫ ∞

−∞

Φ(jΩ)e jΩtdΩ

= . . . see Module 5.5

= GΩNπ

sinc

(

ΩNπ

t

)

6.1 21

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

◮ normalization: G =π

ΩN

◮ total bandwidth: ΩB = 2ΩN

◮ define Ts =2π

ΩB=

π

ΩN

6.1 22

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

Φ(jΩ) =π

ΩNrect

(

Ω

2ΩN

)

ϕ(t) = sinc

(

t

Ts

)

6.1 23

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

0−ΩN ΩN

π/ΩN

6.1 24

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

0 Ts−Ts 2Ts 3Ts 4Ts

0

1

6.1 25

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

END OF MODULE 6.1

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

Module 6.2: InterpolationDigital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

Overview:

◮ Polynomial interpolation

◮ Local interpolation

◮ Sinc interpolation

6.2 26

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

Overview:

6.2 26

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

Overview:

6.2 26

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

Interpolation

x [n] −→ x(t)

“fill the gaps” between samples

6.2 27

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

Example

b

b−1

0

1

2

6.2 28

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

Example

b

b−1

0

1

2

6.2 28

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

Interpolation requirements

◮ decide on Ts

◮ make sure x(nTs) = x [n]

◮ make sure x(t) is smooth

6.2 29

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

◮ decide on Ts

6.2 29

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

◮ decide on Ts

6.2 29

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

Why smoothness?

◮ jumps (1st order discontinuities) would require the signal to move “faster than light”...

◮ 2nd order discontinuities would require infinite acceleration

◮ ...

◮ the interpolation should be infinitely differentiable

◮ “natural” solution: polynomial interpolation

6.2 30

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

Why smoothness?

◮ ...

6.2 30

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

Why smoothness?

◮ ...

6.2 30

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

Why smoothness?

◮ ...

6.2 30

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

Why smoothness?

◮ ...

6.2 30

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

Polynomial interpolation

◮ N points → polynomial of degree (N − 1)

◮ p(t) = a0 + a1t + a2t2 + . . . + aN−1t

(N−1)

◮ “naive” approach:

p(0) = x [0]

p(Ts) = x [1]

p(2Ts) = x [2]

. . .

p((N − 1)Ts) = x [N − 1]

6.2 31

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

◮ p(t) = a0 + a1t + a2t2 + . . . + aN−1t

(N−1)

p(0) = x [0]

p(Ts) = x [1]

p(2Ts) = x [2]

. . .

p((N − 1)Ts) = x [N − 1]

6.2 31

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

◮ p(t) = a0 + a1t + a2t2 + . . . + aN−1t

(N−1)

p(0) = x [0]

p(Ts) = x [1]

p(2Ts) = x [2]

. . .

p((N − 1)Ts) = x [N − 1]

6.2 31

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

Without loss of generality:

◮ consider a symmetric interval IN = [−N, . . . ,N]

◮ set Ts = 1

p(−N) = x [−N]p(−N + 1) = x [−N + 1]

. . .

p(0) = x [0]

. . .

p(N) = x [N]

6.2 32

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

Without loss of generality:

◮ consider a symmetric interval IN = [−N, . . . ,N]

◮ set Ts = 1

p(−N) = x [−N]p(−N + 1) = x [−N + 1]

. . .

p(0) = x [0]

. . .

p(N) = x [N]

6.2 32

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

Lagrange interpolation

◮ PN : space of degree-2N polynomials over IN

◮ a basis for PN is the family of 2N + 1 Lagrange polynomials

L(N)n (t) =

N∏

k=−Nk 6=n

t − kn − k n = −N, . . . ,N

6.2 33

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

Lagrange interpolation polynomials

L2−2(t)

−2 −1 0 1 2

−0.50

0.00

0.50

1.00

1.50

6.2 34

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

L2−2(t)L2−1(t)

−2 −1 0 1 2

−0.50

0.00

0.50

1.00

1.50

6.2 34

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

L2−2(t)L2−1(t)L20(t)

−2 −1 0 1 2

−0.50

0.00

0.50

1.00

1.50

6.2 34

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

L2−2(t)L2−1(t)L20(t)L21(t)

−2 −1 0 1 2

−0.50

0.00

0.50

1.00

1.50

6.2 34

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

L2−2(t)L2−1(t)L20(t)L21(t)L22(t)

−2 −1 0 1 2

−0.50

0.00

0.50

1.00

1.50

6.2 34

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

p(t) =

N∑

n=−N

x [n]L(N)n (t)

6.2 35

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

The Lagrange interpolation is the sought-after polynomial interpolation:

◮ polynomial of degree 2N through 2N + 1 points is unique

◮ the Lagrangian interpolator satisfies

p(n) = x [n] for −N ≤ n ≤ N

since

L(N)n (m) =

{

1 if n = m0 if n 6= m − N ≤ n,m ≤ N

6.2 36

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

b

−2 −1 0 1 2−1

0

1

2

6.2 37

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

x [−2]L(2)−2(t)

b

bb

−2 −1 0 1 2−1

0

1

2

6.2 37

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

x [−1]L(2)−1(t)

b

bb

−2 −1 0 1 2−1

0

1

2

6.2 37

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

x [0]L(2)0 (t)

b

bb

−2 −1 0 1 2−1

0

1

2

6.2 37

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

x [1]L(2)1 (t)

b

bb

−2 −1 0 1 2−1

0

1

2

6.2 37

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

x [2]L(2)2 (t)

b

b bb

−2 −1 0 1 2−1

0

1

2

6.2 37

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

b

−2 −1 0 1 2−1

0

1

2

6.2 37

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

b

−2 −1 0 1 2−1

0

1

2

6.2 37

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

key property:

◮ maximally smooth (infinitely many continuous derivatives)

drawback:

◮ interpolation “bricks” depend on N

6.2 38

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

Relaxing the interpolation requirements

◮ decide on Ts

6.2 39

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

Relaxing the interpolation requirements

◮ decide on Ts

6.2 39

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

Zero-order interpolation

b

−2 −1 0 1 2−1

0

1

2

6.2 40

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

b

−2 −1 0 1 2−1

0

1

2

6.2 40

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

◮ x(t) = x [ ⌊t + 0.5⌋ ], −N ≤ t ≤ N

◮ x(t) =

N∑

n=−N

x [n] rect(t − n)

◮ interpolation kernel: i0(t) = rect(t)

◮ i0(t): “zero-order hold”

◮ interpolator’s support is 1

◮ interpolation is not even continuous

6.2 41

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

◮ x(t) = x [ ⌊t + 0.5⌋ ], −N ≤ t ≤ N

◮ x(t) =

N∑

n=−N

x [n] rect(t − n)

6.2 41

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

◮ x(t) = x [ ⌊t + 0.5⌋ ], −N ≤ t ≤ N

◮ x(t) =

N∑

n=−N

x [n] rect(t − n)

6.2 41

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

◮ x(t) = x [ ⌊t + 0.5⌋ ], −N ≤ t ≤ N

◮ x(t) =

N∑

n=−N

x [n] rect(t − n)

6.2 41

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

◮ x(t) = x [ ⌊t + 0.5⌋ ], −N ≤ t ≤ N

◮ x(t) =

N∑

n=−N

x [n] rect(t − n)

6.2 41

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

◮ x(t) = x [ ⌊t + 0.5⌋ ], −N ≤ t ≤ N

◮ x(t) =

N∑

n=−N

x [n] rect(t − n)

6.2 41

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

b

−2 −1 0 1 2

−1

0

1

2

6.2 42

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

b

bb

−2 −1 0 1 2

−1

0

1

2

6.2 42

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

b

bb

−2 −1 0 1 2

−1

0

1

2

6.2 42

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

b

bb

−2 −1 0 1 2

−1

0

1

2

6.2 42

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

b

bb

−2 −1 0 1 2

−1

0

1

2

6.2 42

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

b

b bb

−2 −1 0 1 2

−1

0

1

2

6.2 42

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

b

−2 −1 0 1 2

−1

0

1

2

6.2 42

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

First-order interpolation

b

−2 −1 0 1 2

−1

0

1

2

6.2 43

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

◮ “connect the dots” strategy

◮ x(t) =

N∑

n=−N

x [n] i1(t − n)

◮ interpolation kernel:

i1(t) =

{

1− |t| |t| ≤ 10 otherwise

◮ interpolation is continuous but derivative is not

6.2 44

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

◮ x(t) =

N∑

n=−N

x [n] i1(t − n)

i1(t) =

{

6.2 44

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

◮ x(t) =

N∑

n=−N

x [n] i1(t − n)

i1(t) =

{

6.2 44

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

◮ x(t) =

N∑

n=−N

x [n] i1(t − n)

i1(t) =

{

6.2 44

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

◮ x(t) =

N∑

n=−N

x [n] i1(t − n)

i1(t) =

{

6.2 44

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

b

−2 −1 0 1 2

−1

0

1

2

6.2 45

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

b

bb

−2 −1 0 1 2

−1

0

1

2

6.2 45

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

b

bb

−2 −1 0 1 2

−1

0

1

2

6.2 45

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

b

bb

−2 −1 0 1 2

−1

0

1

2

6.2 45

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

b

bb

−2 −1 0 1 2

−1

0

1

2

6.2 45

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

b

b bb

−2 −1 0 1 2

−1

0

1

2

6.2 45

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

b

−2 −1 0 1 2

−1

0

1

2

6.2 45

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

Third-order interpolation

◮ x(t) =N∑

n=−N

x [n] i3(t − n)

◮ interpolation kernel obtained by splicing two cubic polynomials

◮ interpolation is continuous up to second derivative

6.2 46

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

◮ x(t) =N∑

n=−N

x [n] i3(t − n)

6.2 46

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

◮ x(t) =N∑

n=−N

x [n] i3(t − n)

6.2 46

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

◮ x(t) =N∑

n=−N

x [n] i3(t − n)

6.2 46

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

b

−2 −1 0 1 2

−1

0

1

2

6.2 47

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

b

bb

−2 −1 0 1 2

−1

0

1

2

6.2 47

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

b

bb

−2 −1 0 1 2

−1

0

1

2

6.2 47

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

b

bb

−2 −1 0 1 2

−1

0

1

2

6.2 47

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

b

bb

−2 −1 0 1 2

−1

0

1

2

6.2 47

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

b

b bb

−2 −1 0 1 2

−1

0

1

2

6.2 47

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

b

−2 −1 0 1 2

−1

0

1

2

6.2 47

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

Local interpolation schemes

x(t) =

N∑

n=−N

x [n] ic(t − n)

Interpolator’s requirements:

◮ ic(0) = 1

◮ ic(t) = 0 for t a nonzero integer.

6.2 48

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

Local interpolation schemes

x(t) =

N∑

n=−N

x [n] ic(t − n)

Interpolator’s requirements:

◮ ic(0) = 1

◮ ic(t) = 0 for t a nonzero integer.

6.2 48

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

Local interpolators

i0(t)

−2 −1 0 1 20

1

6.2 49

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

Local interpolators

i0(t)i1(t)

−2 −1 0 1 20

1

6.2 49

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

Local interpolators

i0(t)i1(t)i3(t)

−2 −1 0 1 20

1

6.2 49

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

Local interpolation

key property:

◮ same interpolating function independently of N

drawback:

◮ lack of smoothness

6.2 50

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

A remarkable result:

limN→∞

L(N)n (t) = sinc (t − n)

in the limit, local and global interpolation are the same!

6.2 51

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

A remarkable result:

limN→∞

L(N)n (t) = sinc (t − n)

in the limit, local and global interpolation are the same!

6.2 51

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

Sinc interpolation formula

x(t) =∞∑

n=−∞x [n] sinc

(

t − nTsTs

)

6.2 52

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

Sinc interpolation

b

bb

b

bb

b

b b

b

00

1

6.2 53

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

Sinc interpolation

b

bb

b

bb

b

b b

bbb

00

1

6.2 53

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

Sinc interpolation

b

bb

b

bb

b

b b

b

bb

00

1

6.2 53

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

Sinc interpolation

b

bb

b

bb

b

b b

b

bb

00

1

6.2 53

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

Sinc interpolation

b

bb

b

bb

b

b b

b

bb

00

1

6.2 53

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

Sinc interpolation

b

bb

b

bb

b

b b

b

bb

00

1

6.2 53

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

Sinc interpolation

b

bb

b

bb

b

b b

b

00

1

6.2 53

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

Sinc interpolation

b

bb

b

bb

b

b b

b

00

1

6.2 53

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

Sinc interpolation

b

bb

b

bb

b

b b

b

00

1

6.2 53

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

“Proof” that L(N)n (t)→ sinc (t − n)

◮ real proof is rather technical (see the book)

◮ intuition: sinc(t − n) and L(∞)n (t) share an infinite number of zeros:

sinc(m − n) = δ[m − n] m, n ∈ Z

L(N)n (m) = δ[m − n] m, n ∈ Z, −N ≤ n,m ≤ N

6.2 54

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

“Proof” that L(N)n (t)→ sinc (t − n)

◮ real proof is rather technical (see the book)

◮ intuition: sinc(t − n) and L(∞)n (t) share an infinite number of zeros:

sinc(m − n) = δ[m − n] m, n ∈ Z

L(N)n (m) = δ[m − n] m, n ∈ Z, −N ≤ n,m ≤ N

6.2 54

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

Graphically

−15 −10 −5 0 5 10 15

0

1

6.2 55

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

Graphically

sinc(t)L1000 (t)

−15 −10 −5 0 5 10 15

0

1

6.2 55

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

Graphically

sinc(t)L2000 (t)

−15 −10 −5 0 5 10 15

0

1

6.2 55

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

Graphically

sinc(t)L3000 (t)

−15 −10 −5 0 5 10 15

0

1

6.2 55

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

END OF MODULE 6.2

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

Module 6.3: The space of bandlimited signalsDigital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

Overview:

◮ Spectrum of interpolated signals

◮ Space of bandlimited functions

◮ Sinc sampling

◮ The sampling theorem

6.3 56

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

Overview:

◮ Sinc sampling

6.3 56

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

Overview:

◮ Sinc sampling

6.3 56

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

Overview:

◮ Sinc sampling

6.3 56

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

Sinc interpolation

the ingredients:

◮ discrete-time signal x [n], n ∈ Z (with DTFT X (e jω))

◮ interpolation interval Ts

◮ the sinc function

the result:

◮ a smooth, continuous-time signal x(t), t ∈ R

what does the spectrum of x(t) look like?

6.3 57

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

Sinc interpolation

the ingredients:

the result:

6.3 57

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

Sinc interpolation

the ingredients:

the result:

6.3 57

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

Key facts about the sinc

ϕ(t) = sinc

(

t

Ts

)

←→ Φ(jΩ) = πΩN

rect

(

Ω

2ΩN

)

Ts =π

ΩNΩN =

π

Ts

6.3 58

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

Key facts about the sinc

0 Ts

0

1ϕ(t)

0 ΩN

0

π/ΩN

Φ(jΩ)

6.3 59

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

Sinc interpolation

x(t) =

∞∑

n=−∞

x [n] sinc

(

t − nTsTs

)

6.3 60

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

Spectral representation (I)

X (jΩ) =

∫ ∞

−∞

x(t) e−jΩtdt

=

∫ ∞

−∞

∞∑

n=−∞

x [n] sinc

(

t − nTsTs

)

e−jΩtdt

=

∞∑

n=−∞

x [n]

∫ ∞

−∞

sinc

(

t − nTsTs

)

e−jΩtdt

=

∞∑

n=−∞

x [n]

(

π

ΩN

)

rect

(

Ω

2ΩN

)

e−jnTsΩ

6.3 61

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

X (jΩ) =

∫ ∞

−∞

x(t) e−jΩtdt

=

∫ ∞

−∞

∞∑

n=−∞

x [n] sinc

(

t − nTsTs

)

e−jΩtdt

=

∞∑

n=−∞

x [n]

∫ ∞

−∞

sinc

(

t − nTsTs

)

e−jΩtdt

=

∞∑

n=−∞

x [n]

(

π

ΩN

)

rect

(

Ω

2ΩN

)

e−jnTsΩ

6.3 61

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

X (jΩ) =

∫ ∞

−∞

x(t) e−jΩtdt

=

∫ ∞

−∞

∞∑

n=−∞

x [n] sinc

(

t − nTsTs

)

e−jΩtdt

=

∞∑

n=−∞

x [n]

∫ ∞

−∞

sinc

(

t − nTsTs

)

e−jΩtdt

=

∞∑

n=−∞

x [n]

(

π

ΩN

)

rect

(

Ω

2ΩN

)

e−jnTsΩ

6.3 61

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

X (jΩ) =

∫ ∞

−∞

x(t) e−jΩtdt

=

∫ ∞

−∞

∞∑

n=−∞

x [n] sinc

(

t − nTsTs

)

e−jΩtdt

=

∞∑

n=−∞

x [n]

∫ ∞

−∞

sinc

(

t − nTsTs

)

e−jΩtdt

=

∞∑

n=−∞

x [n]

(

π

ΩN

)

rect

(

Ω

2ΩN

)

e−jnTsΩ

6.3 61

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

Spectral representation (II)

X (jΩ) =∞∑

n=−∞

x [n]

(

π

ΩN

)

rect

(

Ω

2ΩN

)

e−jnTsΩ

=

(

π

ΩN

)

rect

(

Ω

2ΩN

) ∞∑

n=−∞

x [n]e−j(π/ΩN )Ωn

=

{

(π/ΩN)X (ejπ(Ω/ΩN )) for |Ω| ≤ ΩN

0 otherwise

6.3 62

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

X (jΩ) =∞∑

n=−∞

x [n]

(

π

ΩN

)

rect

(

Ω

2ΩN

)

e−jnTsΩ

=

(

π

ΩN

)

rect

(

Ω

2ΩN

) ∞∑

n=−∞

=

{

0 otherwise

6.3 62

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

X (jΩ) =∞∑

n=−∞

x [n]

(

π

ΩN

)

rect

(

Ω

2ΩN

)

e−jnTsΩ

=

(

π

ΩN

)

rect

(

Ω

2ΩN

) ∞∑

n=−∞

=

{

0 otherwise

6.3 62

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

Spectrum of interpolated signals

−π −π/2 0 π/2 π0

1

X(e

jω)

0ΩN ΩN

T =Ts

X(jΩ)

6.3 63

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

−π −π/2 0 π/2 π0

1

X(e

jω)

0ΩN/2 ΩN/2

T =2Ts

X(jΩ)

6.3 63

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

−π −π/2 0 π/2 π0

1

X(e

jω)

02ΩN 2ΩN

T =Ts/2

X(jΩ)

6.3 63

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

pick interpolation period Ts :

◮ X (jΩ) is ΩN -bandlimited, with ΩN = π/Ts

◮ fast interpolation (Ts small) → wider spectrum

◮ slow interpolation (Ts large) → narrower spectrum

◮ (for those who remember...) it’s like changing the speed of a record player

6.3 64

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

6.3 64

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

6.3 64

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

6.3 64

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

Space of bandlimited functions

Tsx [n] ∈ ℓ2(Z) −−−−−−−−−−−−−−→ x(t) ∈ L2(R)

ΩN -BL

6.3 65

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

Space of bandlimited functions

Tsx [n] ∈ ℓ2(Z) ←−−−−−−−−−−−−−→ x(t) ∈ L2(R)

? ΩN -BL

6.3 65

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

Let’s lighten the notation

for a while we will proceed with

◮ Ts = 1

◮ ΩN = π

(derivations in the general case are in the book)

6.3 66

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

The road to the sampling theorem

claims:

◮ the space of π-bandlimited functions is a Hilbert space

◮ the functions ϕ(n)(t) = sinc(t − n), with n ∈ Z, form a basis for the space

◮ if x(t) is π-BL, the sequence x [n] = x(n), with n ∈ Z, is a sufficient representation(i.e. we can reconstruct x(t) from x [n])

6.3 67

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

claims:

6.3 67

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

claims:

6.3 67

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

The space π-BL

◮ clearly a vector space because π-BL ⊂ L2(R) (and linear combinations of π-BL functionsare π-BL functions)

◮ inner product is standard inner product in L2(R)

◮ completeness... that’s more delicate

6.3 68

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

The space π-BL

6.3 68

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

The space π-BL

6.3 68

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

The space of π-BL functions

recap:

◮ inner product:

〈x(t), y(t)〉 =∫ ∞

−∞

x∗(t)y(t)dt

◮ convolution:(x ∗ y)(t) = 〈x∗(τ), y(t − τ)〉

6.3 69

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

A basis for the π-BL space

ϕ(n)(t) = sinc(t − n), n ∈ Z

〈ϕ(n)(t), ϕ(m)(t)〉 = 〈ϕ(0)(t − n), ϕ(0)(t −m)〉

= 〈ϕ(0)(t − n), ϕ(0)(m − t)〉

=

∫ ∞

−∞

sinc(t − n) sinc(m − t) dt

=

∫ ∞

−∞

sinc(τ) sinc((m − n)− τ) dτ

= (sinc ∗ sinc)(m − n)

6.3 70

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

ϕ(n)(t) = sinc(t − n), n ∈ Z

〈ϕ(n)(t), ϕ(m)(t)〉 = 〈ϕ(0)(t − n), ϕ(0)(t −m)〉

= 〈ϕ(0)(t − n), ϕ(0)(m − t)〉

=

∫ ∞

−∞

=

∫ ∞

−∞

6.3 70

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

ϕ(n)(t) = sinc(t − n), n ∈ Z

〈ϕ(n)(t), ϕ(m)(t)〉 = 〈ϕ(0)(t − n), ϕ(0)(t −m)〉

= 〈ϕ(0)(t − n), ϕ(0)(m − t)〉

=

∫ ∞

−∞

=

∫ ∞

−∞

6.3 70

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

ϕ(n)(t) = sinc(t − n), n ∈ Z

〈ϕ(n)(t), ϕ(m)(t)〉 = 〈ϕ(0)(t − n), ϕ(0)(t −m)〉

= 〈ϕ(0)(t − n), ϕ(0)(m − t)〉

=

∫ ∞

−∞

=

∫ ∞

−∞

6.3 70

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

ϕ(n)(t) = sinc(t − n), n ∈ Z

〈ϕ(n)(t), ϕ(m)(t)〉 = 〈ϕ(0)(t − n), ϕ(0)(t −m)〉

= 〈ϕ(0)(t − n), ϕ(0)(m − t)〉

=

∫ ∞

−∞

=

∫ ∞

−∞

6.3 70

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

now use the convolution theorem knowing that:

FT {sinc(t)} = rect(

Ω

2π

)

(sinc ∗ sinc)(m − n) = 12π

∫ ∞

−∞

[

rect

(

Ω

2π

)]2

e jΩ(m−n)dΩ

=1

2π

∫ π

−πe jΩ(m−n)dΩ

=

{

1 for m = n

0 otherwise

6.3 71

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

Ω

2π

)

∫ ∞

−∞

[

rect

(

Ω

2π

)]2

e jΩ(m−n)dΩ

=1

2π

∫ π

=

{

1 for m = n

0 otherwise

6.3 71

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

Ω

2π

)

∫ ∞

−∞

[

rect

(

Ω

2π

)]2

e jΩ(m−n)dΩ

=1

2π

∫ π

=

{

1 for m = n

0 otherwise

6.3 71

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

Sampling as a basis expansion

for any x(t) ∈ π-BL:

〈ϕ(n)(t), x(t)〉 = 〈sinc(t − n), x(t)〉 = 〈sinc(n − t), x(t)〉= (sinc ∗ x)(n)

=1

2π

∫ ∞

−∞

rect

(

Ω

2π

)

X (jΩ)e jΩndΩ

=1

2π

∫ ∞

−∞

X (jΩ)e jΩndΩ

= x(n)

6.3 72

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

=1

2π

∫ ∞

−∞

rect

(

Ω

2π

)

X (jΩ)e jΩndΩ

=1

2π

∫ ∞

−∞

X (jΩ)e jΩndΩ

= x(n)

6.3 72

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

=1

2π

∫ ∞

−∞

rect

(

Ω

2π

)

X (jΩ)e jΩndΩ

=1

2π

∫ ∞

−∞

X (jΩ)e jΩndΩ

= x(n)

6.3 72

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

=1

2π

∫ ∞

−∞

rect

(

Ω

2π

)

X (jΩ)e jΩndΩ

=1

2π

∫ ∞

−∞

X (jΩ)e jΩndΩ

= x(n)

6.3 72

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

=1

2π

∫ ∞

−∞

rect

(

Ω

2π

)

X (jΩ)e jΩndΩ

=1

2π

∫ ∞

−∞

X (jΩ)e jΩndΩ

= x(n)

6.3 72

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

Sampling as a basis expansion, π-BL

Analysis formula:

x [n] = 〈sinc(t − n), x(t)〉

Synthesis formula:

x(t) =∞∑

n=−∞

x [n] sinc(t − n)

6.3 73

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

Sampling as a basis expansion, ΩN-BL

Analysis formula:

x [n] = 〈sinc(

t − nTsTs

)

, x(t)〉 = Ts x(nTs)

Synthesis formula:

x(t) =1

Ts

∞∑

n=−∞

x [n] sinc

(

t − nTsTs

)

6.3 74

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

The sampling theorem

◮ the space of ΩN -bandlimited functions is a Hilbert space

◮ set Ts = π/ΩN

◮ the functions ϕ(n)(t) = sinc((t − nTs)/Ts) form a basis for the space

◮ for any x(t) ∈ ΩN -BL the coefficients in the sinc basis are the (scaled) samples Ts x(nTs)

for any x(t) ∈ ΩN -BL, a sufficient representation is the sequence x [n] = x(nTs)

6.3 75

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

The sampling theorem

◮ the space of ΩN -bandlimited functions is a Hilbert space

◮ set Ts = π/ΩN

◮ the functions ϕ(n)(t) = sinc((t − nTs)/Ts) form a basis for the space

◮ for any x(t) ∈ ΩN -BL the coefficients in the sinc basis are the (scaled) samples Ts x(nTs)

for any x(t) ∈ ΩN -BL, a sufficient representation is the sequence x [n] = x(nTs)

6.3 75

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

The sampling theorem, corollary

◮ ΩN -BL ⊆ Ω-BL for any Ω ≥ ΩN

for any x(t) ∈ ΩN -BL, a sufficient representation is the sequencex [n] = x(nTs) for any Ts ≤ π/ΩN

6.3 76

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

The sampling theorem, corollary

◮ ΩN -BL ⊆ Ω-BL for any Ω ≥ ΩN

for any x(t) ∈ ΩN -BL, a sufficient representation is the sequencex [n] = x(nTs) for any Ts ≤ π/ΩN

6.3 76

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

The sampling theorem, in hertz

any signal x(t) bandlimited to FN Hz can be sampled with no loss of informationusing a sampling frequency Fs ≥ 2FN (i.e. a sampling period Ts ≤ 1/2FN)

6.3 77

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

END OF MODULE 6.3

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

Module 6.4: Sampling and Aliasing - IntroductionDigital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

Overview:

◮ “Raw” sampling

◮ Sinusoidal aliasing

6.4 78

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

Overview:

◮ “Raw” sampling

◮ Sinusoidal aliasing

6.4 78

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

Sinc Sampling

x [n] = 〈sinc(

t − nTsTs

)

, x(t)〉

6.4 79

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

Sinc Sampling

x [n] = (sincTs ∗x)(nTs )

6.4 79

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

Sinc Sampling

x [n] = (sincTs ∗x)(nTs )

x(t) x [n]

ΩN Ts

6.4 79

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

“Raw” Sampling

x [n] = x(nTs)

x(t) x [n]

Ts

6.4 80

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

Remember the wagonwheel effect?

6.4 81

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

The continuous-time complex exponential

x(t) = e jΩ0t

◮ always periodic, period T = 2π/Ω0

◮ all angular speeds are allowed

◮ FT{

e jΩ0t}

= 2πδ(Ω − Ω0)

◮ bandlimited to Ω0

6.4 82

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

x(t) = e jΩ0t

◮ FT{

e jΩ0t}

= 2πδ(Ω − Ω0)

6.4 82

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

x(t) = e jΩ0t

◮ FT{

e jΩ0t}

= 2πδ(Ω − Ω0)

6.4 82

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

x(t) = e jΩ0t

◮ FT{

e jΩ0t}

= 2πδ(Ω − Ω0)

6.4 82

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

Re

Im

e jΩ0tb

6.4 83

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

Raw samples of the continuous-time complex exponential

x [n] = e jΩ0Tsn

◮ raw samples are snapshots at regular intervals of the rotating point

◮ resulting digital frequency is ω0 = Ω0Ts

6.4 84

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

Raw samples of the continuous-time complex exponential

x [n] = e jΩ0Tsn

◮ raw samples are snapshots at regular intervals of the rotating point

◮ resulting digital frequency is ω0 = Ω0Ts

6.4 84

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

When Ts < π/Ω0, ω0 < π . . .

Re

Im

x[0]b

x[1]b

x[2]b

x[3]b

x[4]b

x[5] b

x[6]b

6.4 85

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

When π/Ω0 < Ts < 2π/Ω0, π < ω0 < 2π . . .

Re

Im

x[0]b

6.4 86

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

When π/Ω0 < Ts < 2π/Ω0, π < ω0 < 2π . . .

Re

Im

b

x[1]bΩ0Ts

6.4 86

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

When π/Ω0 < Ts < 2π/Ω0, π < ω0 < 2π . . .

Re

Im

bb

x[2]b

Ω0Ts

6.4 86

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

When π/Ω0 < Ts < 2π/Ω0, π < ω0 < 2π . . .

Re

Im

bbb

x[3]b

Ω0Ts

6.4 86

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

When π/Ω0 < Ts < 2π/Ω0, π < ω0 < 2π . . .

Re

Im

bbbb

x[4]b

Ω0Ts

6.4 86

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

When π/Ω0 < Ts < 2π/Ω0, π < ω0 < 2π . . .

Re

Im

bbbb

b

x[5]

b

Ω0Ts

6.4 86

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

When Ts > 2π/Ω0, ω0 > 2π . . .

Re

Im

x[0]b

6.4 87

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

When Ts > 2π/Ω0, ω0 > 2π . . .

Re

Im

b

x[1]b

6.4 87

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

When Ts > 2π/Ω0, ω0 > 2π . . .

Re

Im

b

x[2]b

6.4 87

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

When Ts > 2π/Ω0, ω0 > 2π . . .

Re

Im

b

x[3]b

6.4 87

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

When Ts > 2π/Ω0, ω0 > 2π . . .

Re

Im

b

x[4]b

6.4 87

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

When Ts > 2π/Ω0, ω0 > 2π . . .

Re

Im

b

bb

x[5]b

6.4 87

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

Aliasing

x(t) interp x̂(t) =?

Ts Ts

6.4 88

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

Aliasing

x(t) = e jΩ0t

sampling period digital frequency x̂(t)

Ts < π/Ω0 0 < ω0 < π ejΩ0t

π/Ω0 < Ts < 2π/Ω0 π < ω0 < 2π ejΩ1t , Ω1 = Ω0 − 2π/Ts

Ts > 2π/Ω0 ω0 > 2π ejΩ2t , Ω2 = Ω0 mod (2π/Ts )

6.4 89

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

Again, with a simple sinusoid and using hertz

x(t) = cos(2πF0t)

x [n] = x(nTs) = cos(ω0n)

Fs = 1/Ts

ω0 = 2π(F0/Fs)

6.4 90

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

Sampling a Sinusoid

sampling frequency digital frequency result

Fs > 2F0 0 < ω0 < π OKFs = 2F0 ω0 = π max digital frequency: x [n] = (−1)nF0 < Fs < 2F0 π < ω0 < 2π negative frequency ω0 − 2πFs < F0 ω0 > 2π full aliasing: ω0 mod 2π

6.4 91

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

Aliasing: Sampling a Sinusoid

x(t) = cos(6πt) (F0 = 3Hz)

0 1

−1

0

1

6.4 92

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

x(t) = cos(6πt) (F0 = 3Hz)

b b bbbbbbbbbbbbb b b b b

bbbbbbbbbbbbb bb b b b

bbbbbbbbbbbb b b b b

bbbbbbbbbbbb bb b b b

bbbbbbbbbbbb b b b b

bbbbbbbbbbbbb b b

0 1

−1

0

1

Fs = 100Hz

6.4 92

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

x(t) = cos(6πt) (F0 = 3Hz)

b b

b

bb b

b

bb b

b

b b b

b

bb b

b

bb b

b

b b

0 1

−1

0

1

Fs = 50Hz

6.4 92

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

x(t) = cos(6πt) (F0 = 3Hz)

b

0 1

−1

0

1

Fs = 10Hz

6.4 92

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

x(t) = cos(6πt) (F0 = 3Hz)

b

0 1

−1

0

1

Fs = 6Hz

6.4 92

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

x(t) = cos(6πt) (F0 = 3Hz)

b b b

0 1

−1

0

1

Fs = 2.9Hz

6.4 92

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

x(t) = cos(6πt) (F0 = 3Hz)

b b bb

bb

0 1 2

−1

0

1

Fs = 2.9Hz

6.4 92

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

x(t) = cos(6πt) (F0 = 3Hz)

b b bb

bb

b

bb

b

0 1 2 3 4

−1

0

1

Fs = 2.9Hz

6.4 92

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

x(t) = cos(6πt) (F0 = 3Hz)

b b bb

bb

b

bb

b b b bb

bb

b

bb

bb b

b

0 1 2 3 4 5 6 7 8 9 10

−1

0

1

Fs = 2.9Hz

6.4 92

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

END OF MODULE 6.4

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

Module 6.5: Sampling and AliasingDigital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

Overview:

◮ Aliasing for arbitrary spectra

◮ Examples

6.5 93

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

Overview:

◮ Aliasing for arbitrary spectra

◮ Examples

6.5 93

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

Raw-sampling an arbitrary signal

xc(t) x [n] = xc(nTs)

Ts

6.5 94

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

Raw-sampling an arbitrary signal

Xc(jΩ) X (ejω) =?

Ts

6.5 94

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

Key idea

◮ pick Ts (and set ΩN = π/Ts)

◮ pick Ω0 < ΩN

e jΩ0t e jΩ0Tsn

Ts

6.5 95

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

Key idea

e j(Ω0+2ΩN )t e j(Ω0+2ΩN )Tsn

Ts

6.5 95

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

Key idea

e j(Ω0+2ΩN )t e j(Ω0Tsn+2ΩNTsn)

Ts

6.5 95

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

Key idea

e j(Ω0+2ΩN )t e j(Ω0Tsn+2πn)

Ts

6.5 95

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

Key idea

e j(Ω0+2ΩN )t e jΩ0Tsn

Ts

6.5 95

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

Key idea

Ae jΩ0t + Be j(Ω0+2ΩN)t (A+ B)e jΩ0Tsn

Ts

6.5 95

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

Spectrum of raw-sampled signals

start with the inverse Fourier Transform

x [n] = xc(nTs) =1

2π

∫ ∞

−∞

Xc(jΩ)ejΩ nTsdΩ

6.5 96

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

frequencies 2ΩN apart will be aliased, so split the integration interval

x [n] =1

2π

∞∑

k=−∞

∫ (2k+1)ΩN

(2k−1)ΩN

Xc(jΩ)ejΩ nTsdΩ

6.5 97

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

0 ΩN−ΩN 2ΩN−2ΩN 3ΩN−3ΩN 4ΩN−4ΩN

Xc(jΩ)

6.5 98

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

Xc(jΩ)

6.5 98

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

Xc(jΩ)

6.5 98

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

Xc(jΩ)

6.5 98

Digital Sig

nal Proces

sing

Paolo Pran

doni and M

artin Vett

erli

Spectrum of raw-sampled signal

ia802807.us.archive.org · 2018. 8. 12. · the sampling theorem the space of Ω n-bandlimited...

Documents