Digital Signal Processing: Course Review

Course Review References:

Digital Signal Processing: Course Review

Sections: 1.1, 1.2, 1.3, 1.4 2.1, 2.2, 2.3, 2.4, 2.5 3.1, 3.2, 3.3, 3.4 4.1, 4.2, 4.3, 4.4, 5.1, 5.2, 5.4, 5.5 7.1, 7.2, 8.1 11.2, 11.3, 11.4 of John G. Proakis and Dimitris G. Manolakis, Digital Signal Processing: Principles, Algorithms, and Applications, 4th edition, 2007,

Professor Deepa Kundur University of Toronto

and supplementary audio, image and video processing notes.

Professor Deepa Kundur (University of Toronto)Digital Signal Processing: Course Review

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Sampling Theorem If the highest frequency contained in an analog signal xa (t) is Fmax = B and the signal is sampled at a rate Fs > 2Fmax = 2B

Chapter 1: Introduction

then xa (t) can be exactly recovered from its sample values using the interpolation function g (t) =

sin(2πBt) 2πBt

Note: FN = 2B = 2Fmax is called the Nyquist rate.


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Sampling Theorem

Bandlimited Interpolation

Sampling Period = T =

1 1 = Fs Sampling Frequency

Therefore, given the interpolation relation, xa (t) can be written as xa (t) =

x(n) samples

bandlimited interpolation function -- sinc

∞ X

xa (nT )g (t − nT )

n

n=−∞ 0

xa (t) =

∞ X

x(n) g (t − nT )

n=−∞

where xa (nT ) = x(n); called bandlimited interpolation.


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Digital-to-Analog Conversion original/bandlimited interpolated signal

Digital-to-Analog Conversion

x(n)

original/bandlimited interpolated signal

1

n

0

I

I

Common interpolation approaches: bandlimited interpolation, zero-order hold, linear interpolation, higher-order interpolation techniques, e.g., using splines In practice, “cheap” interpolation along with a smoothing filter is employed.


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zero-order hold I

I

1

-3T -2T -T

0

T

2T

t

3T



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Digital-to-Analog Conversion original/bandlimited interpolated signal

-3T -2T -T

I

I

linear interpolation

1

0

T

2T

t

3T

Chapter 2: Discrete-Time Signals and Systems



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Terminology: Input-Output Description


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Classification of Discrete-Time Systems Common System Properties:

input/ excitation

x(n)

Discrete-time System

I

output/ response

Discrete-time signal

Discrete-time signal

I

y(n)

Input-output description (exact structure of system is unknown or ignored): y (n) = T [x(n)] “black box” representation:

vs.

dynamic

time-invariant

vs.

time-variant

linear

vs.

nonlinear

causal

vs.

non-causal

stable

vs.

unstable systems

.. .

T

x(n) −→ y (n) Professor Deepa Kundur (University of Toronto)Digital Signal Processing: Course Review

static

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.. .


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The Convolution Sum

The Convolution Sum

Let the response of a linear time-invariant (LTI) system to the unit sample input δ(n) be h(n). Therefore,

T

δ(n) −→ h(n) T

δ(n − k) −→ h(n − k)

y (n) =

T

α δ(n − k) −→ α h(n − k)

∞ X

x(k)h(n − k) = x(n) ∗ h(n)

k=−∞

T

x(k) δ(n − k) −→ x(k) h(n − k) ∞ ∞ X X T x(k)δ(n − k) −→ x(k)h(n − k) k=−∞

for any LTI system.

k=−∞ T

x(n) −→ y (n)


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Finite vs. Infinite Impulse Response


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System Realization

Implementation: Two classes General expression for Nth-order LCCDE:

Finite impulse response (FIR): y (n) =

M−1 X

N X

) h(k)x(n − k)

nonrecursive systems

k=0

M X ak y (n−k) = bk x(n−k)

a0 , 1

k=0

k=0

Initial conditions: y (−1), y (−2), y (−3), . . . , y (−N). Infinite impulse response (IIR): y (n) =

∞ X

) h(k)x(n − k)

Need: (1) constant scale, (2) addition, (3) delay elements.

recursive systems

k=0


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Building Block Elements

y (n) = −

+

N X

ak y (n − k) +

k=1

Constant multiplier:

+


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Direct Form I IIR Filter Implementation

M X


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Direct Form II IIR Filter Implementation

+

+

+

+

+

+

+

+

+

+

+

LTI All-zero system

+

+

LTI All-pole system

LTI All-pole system

Requires: M + N + 1 multiplications, M + N additions, M + N memory locations Professor Deepa Kundur (University of Toronto)Digital Signal Processing: Course Review

...

+

...

+

...

+

...

+

...

+

...

...

v(n)

k=0

bk x(n − k) nonrecursive | {z } k=0 input 1 N X y (n) = − ak y (n − k) + v (n) recursive |{z} |{z} k=1 output 2 input 2 v (n) = |{z} output 1

Unit advance:

+

bk x(n − k)

is equivalent to the cascade of the following systems:

Unit delay:

Signal multiplier:

M X

...

Adder:

Direct Form I vs. Direct Form II Realizations

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+

LTI All-zero system

Requires: M + N + 1 multiplications, M + N additions, M + N memory locations Professor Deepa Kundur (University of Toronto)Digital Signal Processing: Course Review

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Direct Form II IIR Filter Implementation +

+

+

+

+

+

Chapter 3: The z-Transform and Its Applications

...

...

+

...

+

+

...

+

For N>M

Requires: M + N + 1 multiplications, M + N additions, max(M, N) memory locations Professor Deepa Kundur (University of Toronto)Digital Signal Processing: Course Review

Adder:

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Region of Convergence

Unit delay:

Constant multiplier: Direct z-Transform: ∞Unit advance: X X (z) = x(n)z −n

I

the region of convergence (ROC) of X (z) is the set of all values of z for which X (z) attains a finite value

I

The z-Transform is, therefore, uniquely characterized by:

n=−∞ +

Signal multiplier: I


+

The Direct z-Transform I

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Notation: X (z)

≡

1. expression for X (z) 2. ROC of X (z)

Z{x(n)}

Z

x(n) ←→ X (z)


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ROC Families: Finite Duration Signals


ROC Families: Infinite Duration Signals

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z-Transform Properties Property Notation:


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Convolution using the z-Transform

Linearity: Time shifting:

Time Domain x(n) x1 (n) x2 (n) a1 x1 (n) + a2 x2 (n) x(n − k)

z-Domain X (z) X1 (z) X2 (z) a1 X1 (z) + a2 X2 (z) z −k X (z)

z-Scaling: Time reversal Conjugation: z-Differentiation: Convolution:

an x(n) x(−n) x ∗ (n) n x(n) x1 (n) ∗ x2 (n)

X (a−1 z) X (z −1 ) X ∗ (z ∗ ) dX (z) −z dz X1 (z)X2 (z)

Basic Steps: 1. Compute z-Transform of each of the signals to convolve (time domain → z-domain):

ROC ROC: r2 < |z| < r1 ROC1 ROC2 At least ROC1 ∩ ROC2 At least ROC, except z = 0 (if k > 0) and z = ∞ (if k < 0) |a|r2 < |z| < |a|r1 1 < |z| < r1 r1 2 ROC r2 < |z| < r1 At least ROC1 ∩ ROC2

X1 (z) = Z{x1 (n)} X2 (z) = Z{x2 (n)} 2. Multiply the two z-Transforms (in z-domain): X (z) = X1 (z)X2 (z) 3. Find the inverse z-Transformof the product (z-domain → time domain): x(n) = Z −1 {X (z)}

among others . . .


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Common Transform Pairs 1 2 3 4 5 6

Signal, x(n) δ(n) u(n) an u(n) nan u(n) −an u(−n − 1) −nan u(−n − 1)

7

cos(ω0 n)u(n)

8

sin(ω0 n)u(n)

9

an

10

an

cos(ω0 n)u(n) sin(ω0 n)u(n)

z-Transform, X (z) 1 1 1−z −1 1 1−az −1 az −1 (1−az −1 )2 1 1−az −1 az −1 (1−az −1 )2 1−z −1 cos ω0 1−2z −1 cos ω0 +z −2 z −1 sin ω0 1−2z −1 cos ω0 +z −2 1−az −1 cos ω0 1−2az −1 cos ω0 +a2 z −2 1−az −1 sin ω0 1−2az −1 cos ω0 +a2 z −2

Common Transform Pairs ROC All z |z| > 1 |z| > |a| |z| > |a| |z| < |a| |z| < |a|

1 2 3 4 5 6

Signal, x(n) δ(n) u(n) an u(n) nan u(n) −an u(−n − 1) −nan u(−n − 1)

|z| > 1

7

cos(ω0 n)u(n)

|z| > 1

8 9

cos(ω0 n)u(n)

10

an

sin(ω0 n)u(n)

|z| > |a| |z| > |a|


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Common Transform Pairs

sin(ω0 n)u(n) an

z-Transform, X (z) 1 1 1−z −1 1 1−az −1 az −1 (1−az −1 )2 1 1−az −1 az −1 (1−az −1 )2 1−z −1 cos ω0 1−2z −1 cos ω0 +z −2 z −1 sin ω0 1−2z −1 cos ω0 +z −2 1−az −1 cos ω0 1−2az −1 cos ω0 +a2 z −2 1−az −1 sin ω0 1−2az −1 cos ω0 +a2 z −2

ROC All z |z| > 1 |z| > |a| |z| > |a| |z| < |a| |z| < |a| |z| > 1 |z| > 1 |z| > |a| |z| > |a|


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The System Function Z

h(n) ←→ H(z) I

z-Transform expressions that are a fraction of polynomials in z −1 (or z) are called rational.

Z

time-domain ←→ z-domain Z impulse response ←→ system function Z

y (n) = x(n) ∗ h(n) ←→ Y (z) = X (z) · H(z) I

z-Transforms that are rational represent an important class of signals and systems. Therefore, H(z) =


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Y (z) X (z)


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The System Function of LCCDEs

y (n) = −

N X

ak y (n − k) +

k=1

Z{y (n)} = Z{−

M X

ak y (n − k) +

M X

k=1

Z{y (n)} = − Y (z) = −

k=1 N X

bk x(n − k)

Y (z) +

k=0

N X

N X

The System Function of LCCDEs

"

bk x(n − k)}

ak Z{y (n − k)} +

Y (z) 1 +

k=1

=

bk Z{x(n − k)}

M X

bk z −k X (z)

k=0 N X

# ak z −k

=

k=1

X (z)

M X

bk z −k

k=0

H(z) =

k=0

ak z −k Y (z) +

ak z −k Y (z)

k=1

k=0 M X

M X

N X

Y (z) X (z)

PM

=

−k k=0 bk z h i P 1 + Nk=1 ak z −k

LCCDE ←→ Rational System Function

bk z −k X (z)

k=0

Many signals of practical interest have a rational z-Transform.


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Inversion of the z-Transform Three popular methods: 1. Contour integration: I 1 x(n) = X (z)z n−1 dz 2πj C

Chapter 4: Frequency Analysis of Signals

−1

2. Expansion into a power series in z or z : ∞ X X (z) = x(k)z −k k=−∞

and obtaining x(k) for all k by inspection. 3. Partial-fraction expansion and table look-up.


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CTFT: Intuition

CTFT: Duality

1 x(t) = 2π

Z

∞

X (Ω)e jΩt dΩ

Z ∞ 1 x(t) = X (Ω)e jΩt dΩ 2π −∞ Z ∞ X (Ω) = x(t)e −jΩt dt

−∞

I

We may consider x(t) as a linear combination of e jΩt for Ω ∈ R.

I

The larger |X (Ω)|, the more x(t) will look like a sinusoid with Ω.

−∞

F

Shape A ←→ Shape B F

Shape B ←→ Shape A F

Operation A ←→ Operation B F

Operation B ←→ Operation A


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CTFT: Magnitude and Phase


Complex Sinusoids: Discrete-Time e jωn = cos(ωn) + j sin(ωn)

≡

dst-time complex sinusoid

cos(2πfn)

∞ 1 X (Ω)e jΩt dΩ 2π −∞ Z ∞ 1 = |X (Ω)|e j∠X (Ω) e jΩt dΩ 2π −∞ Z ∞ = |X (Ω)|e j(Ωt+∠X (Ω)) df

Z

x(t)

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=

n

∞ 0

I

|X (Ω)| dictates the relative presence of the sinusoid of frequency Ω in x(t).

I

∠X (Ω) dictates the relative alignment of the sinusoid of frequency Ω in x(t).

sin(2πfn)

n

0


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Classification of Fourier Pairs

Duality PER

PERIODIC

APERIODIC

CTS-TIME Continuous-Time Fourier Series (CTFS) Continuous-Time Fourier Transform (CTFT)

DST-TIME Discrete-Time Fourier Series (DTFS) Discrete-Time Fourier Transform (DTFT)

APER

CTS-TIME P j2πkF0 t x(t) = ∞ k=−∞ ck e R ck = T1p Tp x(t)e −j2πkF0 t dt R∞ 1 jΩt dΩ x(t) = 2π −∞ X (Ω)e R∞ −jΩt X (Ω) = −∞ x(t)e dt

DST-TIME P j2πkn/N x(n) = N−1 k=0 ck e P N−1 ck = N1 n=0 x(n)e −j2πkn/N R 1 X (ω)e jωn dω x(n) = 2π P∞2π X (ω) = n=−∞ x(n)e −jωn

F

periodic ←→ discrete F

discrete ←→ periodic F

aperiodic ←→ continuous F

continuous ←→ aperiodic Professor Deepa Kundur (University of Toronto)Digital Signal Processing: Course Review

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Discrete-Time Fourier Series (DTFS)


DTFS: Example Pair

For discrete-time periodic signals with period N: I

x(n) x(n) A A

Synthesis equation: x(n) =

N−1 X

ck e j2πkn/N

-N -N

k=0 I

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Analysis equation:

-L -L

0 0

L L

N N

n n

cc k k

ck =

N−1 1 X x(n)e −j2πkn/N N n=0

c0 0

0 0

Convergence conditions: None due to finite sums. Professor Deepa Kundur (University of Toronto)Digital Signal Processing: Course Review

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k k

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Discrete-Time Fourier Transform (DTFT)

DTFT: Example Pair

For discrete-time aperiodic signals: I

1 x(n) = 2π I

x(n) x(n)

Synthesis equation: Z

A A

X (ω)e jωn dω

2π -L -L

Analysis equation: X (ω) =

∞ X

0 0

n n

L L

x(n)e −jωn

n=−∞

Convergence conditions: ∞ X

|x(n)| < ∞

0 0

n=−∞ Professor Deepa Kundur (University of Toronto)Digital Signal Processing: Course Review

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DTFT Theorems and Properties ck

x(t)

CTFS

A

c0

sinc -5 -4 -3

t

3 -2 -1

0

1

2

4

5

k

Property Notation:

Linearity: Time shifting: Time reversal Convolution: Correlation:

Time Domain x(n) x1 (n) x2 (n) a1 x1 (n) + a2 x2 (n) x(n − k) x(−n) x1 (n) ∗ x2 (n) rx1 x2 (l) = x1 (l) ∗ x2 (−l)

Wiener-Khintchine:

rxx (l) = x(l) ∗ x(−l)

x(t)

CTFT

A

0

ck

x(n)

DTFS

A

-L

-N

0

X(0)

sinc

t

L

n

N

c0

0

k

Frequency Domain X (ω) X1 (ω) X1 (ω) a1 X1 (ω) + a2 X2 (ω) e −jωk X (ω) X (−ω) X1 (ω)X2 (ω) Sx1 x2 (ω) = X1 (ω)X2 (−ω) = X1 (ω)X2∗ (ω) [if x2 (n) real] Sxx (ω) = |X (ω)|2

x(n)

-L

0

L

among others . . .

DTFT

A n


0

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DTFT Symmetry Properties Time Sequence x(n) x ∗ (n) x ∗ (−n) x(−n) xR (n) jxI (n)

x(n) real

xe0 (n) = 12 [x(n) + x ∗ (−n)] xo0 (n) = 21 [x(n) − x ∗ (−n)]

DTFT X (ω) X ∗ (−ω) X ∗ (ω) X (−ω) Xe (ω) = 12 [X (ω) + X ∗ (−ω)] Xo (ω) = 12 [X (ω) − X ∗ (−ω)] X (ω) = X ∗ (−ω) XR (ω) = XR (−ω) XI (ω) = −XI (−ω) |X (ω)| = |X (−ω)| ∠X (ω) = −∠X (−ω) XR (ω) jXI (ω)


Chapter 5: Frequency Domain Analysis of LTI Systems

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LTI

Linear Time-Invariant LTI (LTI) Systems

Linear Time-Invariant (LTI) Systems

LTI LTI ??? real n

LTI

0

imaginary

LTI

n

LTI Professor Deepa Kundur (University of Toronto)Digital Signal Processing: Course Review

0

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???

Linear Time-Invariant (LTI) Systems

Frequency Response of LTI Systems

LTI

z-Domain

ω-Domain z=e jω

real

H(z) =⇒ H(ω)

real

z=e jω

system function =⇒ frequency response n

n

0

z=e jω

0

imaginary

Y (z) = X (z)H(z) =⇒ Y (ω) = X (ω)H(ω)

imaginary n

n

0

0

imaginary imaginary Professor Deepa Kundur (University of Toronto)Digital Signal Processing: Course Review

53 / 97

complex plane complex plane

LTI


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imaginary

complex plane

imaginary

real

complex plane

imaginary

real

complex plane

LTI Systems as Frequency-Selective Filters

real real

complex plane complex plane

complex plane complex plane complex plane complex plane

complex plane complex plane complex plane

imaginary imaginary

real

I

Filter: device that discriminates, according to some attribute of the input, what passes through it

I

For LTI systems, given Y (ω) = X (ω)H(ω)

real real

imaginary real imaginary imaginary imaginary

real real

I

real imaginary

H(ω) acts as a kind of weighting function or spectral shaping function of the different frequency components of the signal

real imaginary imaginary


LTI system ⇐⇒ Filter

real real

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Ideal Filters

Invertibility of Systems Sharp Transition

Ideal Lowpass Filter

Passband

Stopband

Stopband

I

Invertible system: there is a one-to-one correspondence between its input and output signals

I

the one-to-one nature allows the process of reversing the transformation between input and output; suppose

0 Sharp Transition

Classification: I

lowpass

I

highpass

Passband

Stopband

bandpass

I

bandstop

I

all-pass

Ideal Highpass Filter

0

Sharp Transition Passband

I

Passband

Stopband

y (n) = T [x(n)] where T is one-to-one w (n) = T −1 [y (n)] = T −1 [T [x(n)]] = x(n)

Ideal Bandpass Filter

Passband

0

Passband

Passband Stopband

Passband

Identity System

Ideal Badstop Filter

Stopband

0

Inverse System

Direct System

Ideal All-pass Filter

Passband 0


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Invertibility of LTI Systems


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Invertibility of LTI Systems

Identity System I

Therefore, h(n) ∗ hI (n) = δ(n)

Direct System

Inverse System

I I

Impluse response = LTI

Direct System

LTI

H(z)HI (z) = 1 HI (z) =

Inverse System


For a given h(n), how do we find hI (n)? Consider the z−domain

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1 H(z)


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Invertibility of Rational LTI Systems I

Suppose, H(z) is rational: A(z) B(z) B(z) HI (z) = A(z) poles of H(z) = zeros of HI (z) zeros of H(z) = poles of HI (z)

Chapter 7: The Discrete Fourier Transform

H(z) =


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Intuition


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Intuition Example

aperiodic + dst in time

DTFT ←→

↓ periodic repetition periodic + dst in time

cts + periodic in freq

A

↓ sampling DTFS ←→

dst + periodic in freq 0123 4

-N

one period of dst-time samples n = 0, 1, . . . , N − 1

DFT ←→

N-1 N

...

n

DTFT

one period of dst-freq samples k = 0, 1, . . . , N − 1

Therefore, we define the Discrete Fourier Transform (DFT) as being a computable transform that approximates the DTFT.

0 ...


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Intuition

Intuition

Example

Example

A

-N

0123 4

A

N-1 N

...

n

0123 4

-N

DTFT DTFS

N-1 N

n

DFT

4 -N

...

...

0 1 2 3

N-1 N

4

k

0 1 2 3

-N

N-1N

k

...


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Intuition


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DTFT, DTFS and DFT

Example DTFT

←→

x(n) for all n A

-N

0123 4

...

N-1 N

↓ periodic repetition ∞ X xp (n) = x(n + lN) for all n

n

xˆ(n) where

and 0 1 2 3

N-1N

k

DTFS

←→

X (k) = X (ω)|ω= 2π for all k N k

DFT

ˆ (k) X

ˆ (k) = X

←→

xp (n) 0

for n = 0, . . . , N − 1 otherwise

X (k) 0

for k = 0, . . . , N − 1 otherwise

xˆ(n) =

-N

↓ sampling

l=−∞

DFT

4

X (ω) for all ω

...


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The Discrete Fourier Transform Pair

Frequency Domain Sampling I

I

DFT and inverse-DFT (IDFT): N−1 X

X (k) =

n

x(n)e −j2πk N ,

n=0 N−1 X

1 N

x(n) =

Recall, sampling in time results in a periodic repetition in frequency.

n

X (k)e j2πk N ,

←→

x(n) = xa (t)|t=nT

k = 0, 1, . . . , N − 1 I

n = 0, 1, . . . , N − 1

k=0

∞ 2π 1 X Xa (ω + X (ω) = k) T k=−∞ T

F

Similarly, sampling in frequency results in periodic repetition in time.

Note: we drop the ˆ· notation from now on.

xp (n) =

∞ X

F

←→

x(n + lN)

X (k) = X (ω)|ω= 2π k N

l=−∞


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Frequency Domain Sampling and Reconstruction


N =4

N =4 l=-2

x (n)

x (n+2N)

2

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

n

7

-7

-6

-5

-4

-3

-2

l=1

l=0

+

+

-1

0

1

2

l=2

x (n-N)

x (n)

+

+

...

1

-7

l=-1

x (n+N)

3

4

5

6

l=-2

7

n

-5

-7

-6

-5

-4

-3

-2

-1

0

l=0

l=1

+

2

3

4

5

6

n

7

-7

x (n-2N) +

+ 1

l=2

x (n-N)

x (n)

+

...

1

+

-6

-5

-4

-3

-2

-1

2

3

4

5

6

7

0

1

2

3

4

5

6

7

n

overlap ~

x (n)

support length = 4 = N

-6

x (n+2N)

2

...

no overlap

-7

x (n)

x (n-2N)

l=-1

x (n+N)

-4

-3

-2

-1

0

x~ (n)

support length = 6 > N

2

2

1

1

1

2

3

4

5

6


= x (n)

7

n

-7

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-6

-5

-4

-3

-2

-1

0

1


=x(n)

n 72 / 97


Important DFT Properties

N =4 x(n)

-7

-6

-5

-4

-3

-2

-1

xp(n)

no temporal aliasing

2

2

1

1

0

1

2

3

4

5

6

7

n

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

n

4

5

6

7

n

=x(n) x(n)

-7

-6

-5

-4

-3

-2

-1

0

time-domain aliasing

xp(n)

2

2

1

1

1

2

3

4

5

6

7

n

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

Property Notation: Periodicity: Linearity: Time reversal Circular time shift: Circular frequency shift: Complex conjugate: Circular convolution: Multiplication: Parseval’s theorem:

Time Domain x(n) x(n) = x(n + N) a1 x1 (n) + a2 x2 (n) x(N − n) x((n − l))N x(n)e j2πln/N x ∗ (n) x1 (n) ⊗ x2 (n) x1 (n)x2 (n) PN−1 ∗ n=0 x(n)y (n)

Frequency Domain X (k) X (k) = X (k + N) a1 X1 (k) + a2 X2 (k) X (N − k) X (k)e −j2πkl/N X ((k − l))N X ∗ (N − k) X1 (k)X2 (k) 1 X1 (k) ⊗ X2 (k) N P N−1 1 ∗ k=0 X (k)Y (k) N

=x(n) Professor Deepa Kundur (University of Toronto)Digital Signal Processing: Course Review

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Radix-2 FFT

Chapter 8: The Fast Fourier Transform

Two strategies: I Decimation in time (our focus in the lecture) I Decimation in frequency

I


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Note: We assume that N is a power of two; i.e., N = 2r .


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Radix-2 FFT: Decimation-in-time

Radix-2 FFT: Decimation-in-time

For N = 8.

For N = 8.

x(0) x(4) x(2) x(6)

Combine 2-point DFTs

2-point DFT

x(1) x(5)


2-point DFT

x(3) x(7)

Stage 1

x(0)

2-point DFT


2-point DFT

x(4)

X(0) X(1) X(2) X(3) X(4) X(5) X(6) X(7)

Stage 3

X(0) X(1)

-1 0 W8

x(2)

X(2)

-1 2

x(6)

W8 0

W8

-1

x(5)

0 W8

-1 1 W8 0

W8

-1

-1 0

W8 -1

2 W8 0

W8

-1

X(4) X(5)

2

W8

x(3)

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X(3)

-1

x(1)

x(7)


0

W8

Stage 2

-1 3 W8

-1

-1

X(6) X(7)


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FFT Complexity A = a + W Nr b

a r

b

I

I

A = a - W Nr b

Chapter 11: Multirate Digital Signal Processing

one complex multiplication two complex additions

In total, there are: I I

I

-1

Each butterfly requires: I

I

WN

N 2

butterflies per stage log N stages

Order of the overall DFT computation is: O(N log N).


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80 / 97

Sampling Rate Conversion I

Sampling of Discrete-Time Signals Suppose a discrete-time signal x(n) is sampled by taking every Dth sample as follows:

Goal: Given a discrete-time signal x(n) sampled at period T from an underlying continuous-time signal xa (t), determine a new sequence xˆ(n) that is a sampled version of xa (t) at a different sampling rate Td . x(n) = xa (nT )

xd (n) = x(nD),

xˆ(n) = xa (nTd )

Decimation example: D = 2: x(n)

x(n)

-2

-1

0

1

2

-3

n

3

-2

0

-3

2

n

3

1

2


Td = 2T

-2

n

3 0

1

1

Td

1

-1

-1

xd (n)

x(n)

-2

T

1

T

1

-3

for all n

81 / 97

0

1

3

x (n)

Professor Deepa Kundur (University of Toronto)Digital Signal dProcessing: Course Review

1

-2

2

1/T ...

-1

0

1

2

n

3

1

...

F

0

Xd (F)

xd (n)

1/Td

Td = 2T ...

-2

n

2 -1

-3

0

1

3

1

...

F

0

Xd (F)

xd (n)

1/Td

Td = 4T ...

-1 -3

-2

3

X (F) T

1

-2

n

1 0

Decimation example: D = 2, 4:

x(n)

-3

82 / 97

Td = 4T

-1 -3

Decimation example: D = 2, 4:

n

2 -1

-3

1 0


2

3

n 83 / 97

... 0


F 84 / 97

Downsampling with Anti-Alaising Filter

−π/D ≤ ω ≤ π/D is expanded into −π ≤ ω ≤ π

Decimator LTI Filter

Downsampler

ANTI-ALIASED VERSION

X( ) 1/T

...

... 0

Xd ( ) 1/Td ...

The anti-aliasing filter Hd (ω) should have effective 1 continuous-time frequency cutoff of F0 = 2DT Hz, which is equivalent to a normalized cutoff of:

I

Interpolator

Upsampler

f0 =

LTI 1 Filter 1

F0 1 = · = Fs 2DT Fs 2D

LTI Filter

or

... 0

Xd ( )

Decimator 1Downsampler π

ω0 = 2π

2D

=

1/Td ...

D

... 0


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Interpolation of Discrete-time Signals

−π/D ≤ ω ≤ π/D is expanded into −π ≤ ω ≤ π ANTI-ALIASED VERSION


X( )

...

To achieve this, consider a two-stage process: I Stage 1: Upsample to appropriately compress the spectrum. I Stage 2: Then filter with an appropriate lowpass filter.

... 0

I

We will consider upsampling by a factor of I . I

Xd ( ) ...

Note: we change here the interpolation factor from D to I to distinguish our results from decimation.

... 0


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88 / 97

Interpolation by a Factor I

Interpolation of Discrete-time Signals

Interpolator LTI Filter

Upsampler

I

I

Upsampling (without filtering) can be represented as: x(m/I ) m = 0, ±I , ±2I , . . . v (m) = 0 otherwise V (ω) = X (ωI )

Interpolation only increases the visible resolution of the signal. No new information is gained. Interpolator LTI Filter

LTI Filter

Decimator Downsampler


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Interpolation example: I = 4: upsampling + lowpass filtering


90 / 97

Interpolation example: I = 4: upsampling + lowpass filtering X( )

x(n) 1

... -1 -3

1

-2

2

0

3

n

... 0

V( )

v(n) 1

... -3

-2

-1

0

1

2

3

n

... 0

Y( )

y(n) 1

... -3

-2

-1

0

1

2

3


n 91 / 97

... 0


92 / 97

Sampling Rate Conversion by I /D

−π ≤ ω ≤ π is compressed into −π/I ≤ ω ≤ π/I

Interpolator

X( )

Upsampler

...

LTI Filter

LTI Filter


... 0

I I

x(n): original samples at sampling rate Fx y (n): new samples at sampling rate Fy

XI ( ) ...

... 0


93 / 97


94 / 97

Sampling Rate Conversion by I /D Interpolator Upsampler

LTI Filter

Upsampler

LTI Filter

LTI Filter


Downsampler

H(ω) = Hu (ω)Hd (ω) =


95 / 97

I 0 ≤ |ω| ≤ min(π/D, π/I ) 0 otherwise


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Sampling Rate Conversion by I /D Interpolator Upsampler

LTI Filter

Upsampler

LTI Filter

I/D Rate Converter LTI Filter


Downsampler


97 / 97

Digital Signal Processing: Course Review

Digital Signal Processing: Course Review

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