Up- Up -Sampler Sampler

(1)

1

Multirate

Multirate Digital Signal Digital Signal Processing

Processing

Basic Sampling Rate Alteration Devices

• Up-sampler-Used to increase the sampling rate by an integer factor

• Down-sampler-Used to decrease the sampling rate by an integer factor

2

Up- Up -Sampler Sampler

Time-Domain Characterization

• An up-sampler with an up-sampling factor L, where Lis a positive integer, develops an output sequence with a sampling rate that is Ltimes larger than that of the input sequencex[n]

• Block-diagram representation ] [n x_u

L

x[n] x_u[n]

3

Up- Up -Sampler Sampler

• Up-sampling operation is implemented by inserting equidistant zero-valued samples between two consecutive samples ofx[n]

• Input-output relation

−1 L

⎩⎨⎧ = ± ±

= 0, otherwise

, 2 , , 0 ], / ] [

[ xn L n L LL

n x_u

4

Up- Up -Sampler Sampler

• Figure below shows the up-sampling by a factor of 3of a sinusoidal sequence with a frequency of 0.12Hz obtained using Program 13_1

0 10 20 30 40 50

-1 -0.5 0 0.5 1

Input Sequence

Time index n

Amplitude

0 10 20 30 40 50

-1 -0.5 0 0.5 1

Output sequence up-sampled by 3

Time index n

Amplitude

2 zero-valued samples

5

Up- Up -Sampler Sampler

• In practice, the zero-valued samples

inserted by the up-sampler are replaced with appropriate nonzero values using some type of filtering process

• Process is called interpolationand will be discussed later

6

Down- Down -Sampler Sampler

Time-Domain Characterization

• An down-sampler with a down-sampling factorM, where Mis a positive integer, develops an output sequence y[n]with a sampling rate that is (1/M)-th of that of the input sequencex[n]

• Block-diagram representation

M

x[n] y[n]

(2)

7

Down- Down -Sampler Sampler

• Down-sampling operation is implemented by keeping every M-th sample of x[n]and removing in-between samples to generatey[n]

• Input-output relation y[n] = x[nM]

−1 M

8

Down- Down -Sampler Sampler

• Figure below shows the down-sampling by a factor of 3of a sinusoidal sequence of frequency 0.042Hz obtained using Program 13_2

0 10 20 30 40 50

-1 -0.5 0 0.5 1

Input Sequence

Time index n

Amplitude

0 10 20 30 40 50

-1 -0.5 0 0.5 1

Output sequence down-sampled by 3

Amplitude

Time index n Removed

9

Basic Sampling Rate Basic Sampling Rate

Alteration Devices Alteration Devices

• Sampling periods have not been explicitly shown in the block-diagram representations of the up-sampler and the down-sampler

• This is for simplicity and the fact that the mathematical theory of multirate systems can be understood without bringing the sampling period Tor the sampling frequency into the pictureF_T

10

Down- Down -Sampler Sampler

• Figure below shows explicitly the input- output sampling rates of the down-sampler

M y[n]=x_a(nMT) )

( ] [n x nT x = _a Input sampling frequency

F_T T1

=

Output sampling frequency ' ' 1

T M F_T =F^T =

Up- Up -Sampler Sampler

• Figure below shows explicitly the input- output sampling rates of the up-sampler

Input sampling frequency

F_T T1

=

⎩⎨

⎧ = ± ±

= 0 otherwise

, 2 , , 0 ), /

(nT L n L LK

x_a ) L

( ] [n x nT

x = _a y[n]

Output sampling frequency ' ' 1

LF T F_T = _T =

A Simple

A Simple Multirate Multirate Structure Structure

• The operation of the above multirate structure can be analyzed by writing down the relations between various signal variables, and the input as shown in the next slide

2 2

2 2 +

−1 1 z

z−

] [n x

] [n y ]

[n v

] [n

w w_u[n] ] [n v_u

(3)

13

A Simple

A Simple Multirate Multirate Structure Structure

] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ : ] [

:

8 7 6 5 4 3 2 1 0

x x x x x x x x x n x

n

14

A Simple

A Simple Multirate Multirate Structure Structure

] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ : ] [

:

15 13 11 9 7 5 3 1 1

16 14 12 10 8 6 4 2 0

8 7 6 5 4 3 2 1 0

x x x x x x x x x n w

x x x x x x x x x n v

n

−

15

A Simple

A Simple Multirate Multirate Structure Structure

] [ ]

[ ]

[ ] [ ] [ : ] [

] [ ]

[ ]

[ ] [ ] [ : ] [

] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ : ] [

:

7 0 5 0 3 0 1 0 1

8 0 6 0 4 0 2 0 0

15 13 11 9 7 5 3 1 1

16 14 12 10 8 6 4 2 0

8 7 6 5 4 3 2 1 0

x x

x un w

x x

x un v

n

−

16

A Simple

A Simple Multirate Multirate Structure Structure

] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ : ] [

] [ ] [ ] [ ] [ : ] [

] [ ] [ ] [ ] [ ] [ : ] [

] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ : ] [

:

7 6 5 4 3 2 1 0 1

0 6 0 4 0 2 0 0 0 1

7 0 5 0 3 0 1 0 1

8 0 6 0 4 0 2 0 0

15 13 11 9 7 5 3 1 1

16 14 12 10 8 6 4 2 0

8 7 6 5 4 3 2 1 0

x x x x x x x x x n y

x x

x x n

v

x x

x x x

n w

x x

x x x

n v

n

u u u

−

− −

−

] [ ] [ ] [ ]

[n =v n−1 +w n =xn−1

y _u _u

17

Basic Sampling Rate Basic Sampling Rate

Alteration Devices Alteration Devices

• The up-sampler and the down-sampler are linear but time-varying discrete-time systems

• We illustrate the time-varying property of a down-sampler

• The time-varying property of an up-sampler can be proved in a similar manner

18

Basic Sampling Rate Basic Sampling Rate

Alteration Devices Alteration Devices

• Consider a factor-of-Mdown-sampler defined by

• Its output for an input is then given by

• From the input-output relation of the down- sampler we obtain

y[n] = x[nM]

]

1[n

y x₁[n]=x[n−n₀] ]

[ ] [ ]

[ ₁ ₀

1n x Mn xMn n

y = = −

)]

( [ ]

[n n₀ xM n n₀

y − = −

] [ ] [Mn Mn₀ y₁n

x − ≠

=

(4)

19

Up

Up- -Sampler Sampler

Frequency-Domain Characterization

• Consider first a factor-of-2up-sampler whose input-output relation in the time- domain is given by

⎩⎨⎧ = ± ±

= , otherwise

, , , ], / ] [

[ 0

4 2 0

2 n K

n n x x_u

20

Up- Up -Sampler Sampler

• In terms of the z-transform, the input-output relation is then given by

∑

^∞

−∞

=

∞ −

−∞

=

− =

=

even

] / [ ]

[ )

(

nn

n n

n u

u z x n z xn z

X 2

) ( ]

[m z ² X z² x

m

m=

= ∑^∞

−∞

=

−

21

Up- Up -Sampler Sampler

• In a similar manner, we can show that for a factor-of-Lup-sampler

• On the unit circle, for , the input- output relation is given by

) ( )

( ^L

u z X z

X =

=e^jω

z

) ( )

( ^j ^j ^L

u e X e

X ^ω = ^ω

22

Up

Up- -Sampler Sampler

• Figure below shows the relation between and for L= 2in the case of a typical sequencex[n]

) (e^j^ω

X X_u(e^j^ω)

Up- Up -Sampler Sampler

• As can be seen, a factor-of-2sampling rate expansion leads to a compression of by a factor of 2and a 2-fold repetition in the baseband[0, 2π]

• This process is called imagingas we get an additional “image” of the input spectrum

) (e^j^ω X

Up- Up -Sampler Sampler

• Similarly in the case of a factor-of-L sampling rate expansion, there will be additional images of the input spectrum in the baseband

• Lowpass filtering of removes the images and in effect “fills in” the zero- valued samples in with interpolated sample values

−1 L

−1 L ]

x_u[n ] [n x_u

(5)

25

Up- Up -Sampler Sampler

• Program 13_3can be used to illustrate the frequency-domain properties of the up- sampler shown below for L= 4

0 0.2 0.4 0.6 0.8 1

ω/π

Magnitude

Output spectrum

0 0.2 0.4 0.6 0.8 1

ω/π

Magnitude

Input spectrum

26

Down- Down -Sampler Sampler

Frequency-Domain Characterization

• Applying the z-transform to the input-output relation of a factor-of-Mdown-sampler

we get

• The expression on the right-hand side cannot be directly expressed in terms of X(z)

∑

^∞

−∞

=

= − n

z n

Mn x z

Y( ) [ ] ] [ ] [n xMn y =

27

Down- Down -Sampler Sampler

• To get around this problem, define a new sequence :

• Then

⎩⎨⎧ = ± ±

= , otherwise

, , , ], ] [

int[ 0

2

0 M M K

n n n x x

]

int[n x

∑

^∞

−∞

=

∞ −

−∞

=

− =

=

n

n n

n x Mn z

z Mn x z

Y( ) [ ] _int[ ] ) ( ]

[ ^/ _int ^/

int M

k

M

k X z

z k

x = ¹

=

∑

^∞

−∞

=

−

28

Down- Down -Sampler Sampler

• Now, can be formally related to x[n]

through

where

• A convenient representation of c[n]is given by

where ]

int[n x

] [ ] [ ]

int[n cn xn

x = ⋅

⎩⎨

⎧ = ± ±

= , otherwise

, , , ] ,

[ 0

2 0

1 n M M K

n c

∑

⁻

=

= ¹

0

1 ^M

k kn

WM

n M c[ ]

M j

M e

W = ⁻ ²^π^/

29

Down

Down- -Sampler Sampler

• Taking the z-transform of and making use of

we arrive at

] [ ] [ ]

int[n cn xn

x = ⋅

∑

⁻

=

= ¹

0

1 ^M

k kn

WM

n M c[ ]

n n

M k

kn M n

n W xn z

z M n x n c z

X ⁻

∞

−∞

=

−

=

∞

−∞

=

−

∑ ∑

∑

_⎟⎟

⎠

⎜⎜ ⎞

⎝

= ⎛

= [ ] [ ] [ ]

)

int(

1 0

1

( )

∑

∑ ∑ ⁻

=

− −

=

∞

−∞

=

− ⎟⎟⎠=

⎜⎜ ⎞

⎝

= ⎛ ¹

0 1

0

1

1 ^M

k

k M M

k n

n kn

M X zW

z M W n M x[ ]

30

Down- Down -Sampler Sampler

• Hence,

• On the unit circle, ) ( )

(z X_int z ^/^M

Y = ¹

= ∑⁻

= 1 − 0 1M 1

k

k M M M X(z ^/ W ⁾)

∑

⁻

=

π

− ω

ω = ¹

0

/ ) 2

( )

1 ( ) (

M k

M k j

j X e

e M Y

(6)

31

Down- Down -Sampler Sampler

• Consider a factor-of-2down-sampler with an input x[n]whose spectrum is as shown below

• The DTFTs of the output and the input sequences of this down-sampler are then related as

)}

( ) ( 2{ ) 1

(e^j^ω = X e^j^ω^/² +X −e^j^ω^/² Y

32

Down- Down -Sampler Sampler

• Now implying that the second term in the previous equation is simply obtained by shifting the first term to the right by an amount 2πas shown below

) (

)

(−e^j^ω^/² =X e^j⁽^ω⁻²^π⁾^/² X

) (−e^j^ω^/² X

) (e^j^ω^/² X

33

Down- Down -Sampler Sampler

• The plots of the two terms have an overlap, and hence, in general, the original “shape”

of is lost when x[n]is down- sampled as indicated below

) (e^j^ω X

34

Down- Down -Sampler Sampler

• This overlap causes the aliasingthat takes place due to under-sampling

• There is no overlap, i.e., no aliasing, only if

• Note: is indeed periodic with a period2π, even though the stretched version of is periodic with a period4π

2 / 0

)

(e^j^ω = for ω≥π X

) (e^j^ω X

) (e^j^ω Y

Down- Down -Sampler Sampler

• For the general case, the relation between the DTFTs of the output and the input of a factor-of-Mdown-sampler is given by

• is a sum of Muniformly

shifted and stretched versions of and scaled by a factor of1/M

∑

⁻

=

π

− ω

ω = ¹

0

/ ) 2

( )

1 ( ) (

M k

M k j

j X e

e M Y

) (e^j^ω Y

) (e^j^ω X

Down- Down -Sampler Sampler

• Aliasing is absent if and only if

as shown below for M= 2 2 / for 0 )

(e^j^ω = ω≥π X

M for

e

X( ^j^ω)=0 ω≥π/

(7)

37

Down- Down -Sampler Sampler

• Program 13_4can be used to illustrate the frequency-domain properties of the down- sampler shown below for M= 2

0 0.2 0.4 0.6 0.8 1

ω/π

Magnitude

Input spectrum

0 0.2 0.4 0.6 0.8 1

0 0.1 0.2 0.3 0.4 0.5

ω/π

Magnitude

Output spectrum

38

Down- Down -Sampler Sampler

• The input and output spectra of a down- sampler withM= 3 obtained using Program 13_4are shown below

• Effect of aliasing can be clearly seen

0 0.2 0.4 0.6 0.8 1

ω/π

Magnitude

Input spectrum

0 0.2 0.4 0.6 0.8 1

0 0.1 0.2 0.3 0.4 0.5

ω/π

Magnitude

Output spectrum

39

Cascade Equivalences Cascade Equivalences

• A complex multirate system is formed by an interconnection of the up-sampler, the down-sampler, and the components of an LTI digital filter

• In many applications these devices appear in a cascade form

• An interchange of the positions of the branches in a cascade often can lead to a computationally efficient realization

40

Up

Up- -Sampler and Sampler and Down- Down -sampler Cascade sampler Cascade

• To implement a fractional change in the sampling rate we need to employ a cascade of an up-sampler and a down-sampler

• Consider the two cascade connections shown below

M ] L

[n

x y₁[n]

M L

] [n

x y₂[n]

] [n v₁

] [n v₂

41

Up- Up -Sampler and Sampler and Down- Down -sampler Cascade sampler Cascade

• Consider the top cascade shown in the previous slide

• Here, we have and

• Combining the last two equations we get )

( ) (z X z^L V₁ =

) (

)

( ∑− ^/

=

= ¹ − 0 1 1

1 ^M

k

k M MW z M V

z Y

) (

)

( ∑− ^/

=

= ¹ − 1 0

1 ^M

k

kL M M

L W

z M X

z Y

42

Up- Up -Sampler and Sampler and Down- Down -sampler Cascade sampler Cascade

• We next consider the bottom cascade shown in Slide 36

• Here, we have

and

• Combining the last two equations we get ) (

)

( ∑− ^/

=

= ¹ − 0 2 1

1 ^M

k

k M MW z M X

z V

) ( ) (z V z^L Y₂ = ₂

) (

)

( ∑− ^/

=

= ¹ − 2 0

1 ^M

k

k M M

L W

z M X

z Y

(8)

43

Up- Up -Sampler and Sampler and Down- Down -sampler Cascade sampler Cascade

• It follows from the above that if

• The above equality holds if and only if M and Lare relatively prime, i.e. Mand Ldo not have a common factor that is an integer r> 1, as then and take the same set of values for

) ( ) (z Y z Y₁ = ₂

) (

)

( ^/ ∑− ^/

=

∑− −

=

− = ¹

0 1 1

0

1 M

k

k M M L M

M k

kL M M L

M X z W X z W

k

WM⁻ W_M⁻^kL 1 1

0 −

= M

k , ,K,

44

Noble Identities Noble Identities

• Two other cascade equivalences are shown below

] L [n

x H(z^L) y₂[n] ] L [n

x H(z) y₂[n]

≡

] M [n

x H(z) y₁[n] ] M [n

x H(z^M) y₁[n]

≡

Cascade equivalence #1

Cascade equivalence #2

45

Multirate

Multirate Structures for Structures for Sampling Rate Conversion Sampling Rate Conversion

• From the sampling theorem it is known that the sampling rate of a critically sampled discrete-time signal with a spectrum occupying the full Nyquist range cannot be reduced any further since such a reduction will introduce aliasing

• Hence, the bandwidth of a critically sampled signal must be reduced by lowpass filteringbefore its sampling rate is reduced by a down-sampler

46

Multirate

Multirate Structures for Structures for Sampling Rate Conversion Sampling Rate Conversion

• Likewise, the zero-valued samples introduced by an up-sampler must be interpolated to more appropriate values for an effective sampling rate increase

• We shall show shortly that this interpolation can be achieved simply by digital lowpass filtering

Multirate

Multirate Structures for Structures for Sampling Rate Conversion Sampling Rate Conversion

• Since a fractional-rate sampling rate converter with a rational conversion factor can be realized by cascading an interpolator with a decimator, filters are also needed in the design of such multirate systems

Basic Structures Basic Structures

• Since up-samplingby an integer factor L causes periodic repetition of the basic spectrum, the basic interpolator structure for integer-valued sampling rate increase consists of an up-samplerfollowed by a low-pass filter with a cutoff at as indicated below:

) (z

H π/L

] L [n

x x_u[n] H(z) y[n]

(9)

49

Basic Structures Basic Structures

• The lowpass filter , called the interpolation filter, removes the unwanted imagesin the spectra of the up-sampled signal

• On the other hand, down-sampling by an integer factor Mmay result in aliasing

) (z H

] [n x_u

50

Basic Structures Basic Structures

• Hence, the basic decimator structure for integer-valued sampling rate decrease consists of a lowpss filter with a cutoff atπ/M, followed by the down-sampler as shown below

) (z H

] M [n

x H(z) y[n]

51

Basic Structures Basic Structures

• Here, the lowpass filter , called the decimation filter, bandlimits the input signal x[n]to prior to down-sampling, to ensure no aliasing

• It can be shown that the transpose of a factor-of-Mdecimator is a factor-of-M interpolator

) (z H M

π/

<

ω

52

Basic Structures Basic Structures

• A fractional changein the sampling rate by a rational factor L/Mcan be achieved by cascading a factor-of-Linterpolatorwith a factor-of-Mdecimator

• The interpolator must precede the decimator as shown below to ensure that the baseband ofw[n]is greater than or equal to that of x[n]ory[n]

) M (z

H_d y[n]

L ] [n

x H_u(z)w[n]

53

Basic Structures Basic Structures

• As both the interpolation filter and the decimation filter operate at the same sampling rate, they can be replaced with a single filterdesigned to avoid aliasing that may be caused by down- sampling and eliminate images resulting from up-sampling

) (z H_d

) (z H_u

M ] L

[n

x H(z) y[n]

54

Input

Input- -Output Relation of the Output Relation of the Decimator

Decimator

• For the decimator structure shown below, let h[n]denote the impulse response of the decimation filter H(z)

• Then and

] M [n

x H(z) v[n] y[n]

∑ −

= ^∞

−∞

l= [ l] [l] ]

[n hn x

v

] [ ] [n vMn y =

(10)

55

Input

Input- -Output Relation of the Output Relation of the Decimator

Decimator

• Combining the last two equations we arrive at the desired input-output relation of the decimator given by

• In the z-domain, the input-output relation of the decimation filter is given by

∑ −

= ^∞

−∞

l= [ l] [l] ]

[n h Mn x

y

) ( ) ( )

(z H z X z

V =

56

Input

Input- -Output Relation of the Output Relation of the Decimator

Decimator

• Now the input-output relation of the dowm- sampler is given by

• Combining the last two equations we arrive at the input-output relation of the decimator as

= ∑⁻

= 1 − 0

1 ^M 1 k

k M MW z M V

z

Y( ) ( ^/ )

) (

)

( ^M ^/ ^/^M _M^k

k

k M

MW X z W

z M H

z

Y ⁻ ⁻

=

∑ −

= ¹ ¹

0

1 1

57

Input

Input- -Output Relation of the Output Relation of the Interpolator

Interpolator

• For the interpolator structure shown below, let h[n]denote the impulse response of the decimation filter H(z)

• Then and

] L [n

x x_u[n] H(z) y[n]

∑ −

= ^∞

−∞

l= [ l] [l] ]

[n hn x_u

y

K , , , ],

[ ]

[Lm =xm m=0±1 ±2 x_u

58

Input

Input- -Output Relation of the Output Relation of the Interpolator

Interpolator

• Combining the last two equations and making a change of a variable, we arrive at the desired time-domain input-output relation of the interpolator as

• In the z-domain, the input-output relation of the interpolator is thus given by

∑ −

= ^∞

−∞

= m

m x Lm n h n

y[ ] [ ] [ ]

) ( ) ( )

(z H z X z^L Y =

Input

Input- -Output Relation of the Output Relation of the Fractional

Fractional -Rate Converter - Rate Converter

• Here, in the time-domain the input-output relation is given by

• In the z-domain it is given by

∑ −

= ^∞

−∞

= m

m x Lm Mn h n

y[ ] [ ] [ ]

) (

)

( ^/ ^L^/^M _M^kL

M k

k M

MW X z W

z M H

z

Y ⁻

−

=

∑ −

= ¹

0

1 1

Interpolation Filter Interpolation Filter

Specifications Specifications

• Assume x[n]has been obtained by sampling a continuous-time signal at the Nyquist rate

• If and denote the Fourier transforms of and x[n], respectively, then it can be shown

• where is the sampling period ) (t x_a

) (t x_a ) (jΩ

X_a X(e^j^ω)

⎟⎠

⎜ ⎞

⎝

⎛ ω− π

= ∑^∞

−∞

= ω

o o

)

( T

k j X j

e T X

k a

j 1 2

To

(11)

61

Interpolation Filter Interpolation Filter

Specifications Specifications

• Figures below show andx[n] obtained by sampling at the Nyquist rate

) (t x_a

] [n x

) (t x_a ) (t x_a

62

Interpolation Filter Interpolation Filter

Specifications Specifications

• Figures below show the Fourier transforms of and x[n]

) (t x_a

) (jΩ X_a

63

Interpolation Filter Interpolation Filter

Specifications Specifications

• Since the sampling is being performed at the Nyquist rate, there is no overlap between the shifted spectras of

• If we instead sample at a much higher rate yielding y[n], its Fourier transform is related to through

) / (j T_o X ω

) (t x_a L

T T= _o/

) (e^j^ω

Y X_a(jΩ)

∑ ⎟

⎠

⎜ ⎞

⎝

⎛ ω− π

∑ ⎟⎠=

⎜ ⎞

⎝

⎛ ω− π

= ^∞

−∞

=

∞

−∞

= ω

k a

j

L T

k j X j T

L T

k j X j e T

Y( ) /

o o

2 2

1

64

Interpolation Filter Interpolation Filter

Specifications Specifications

• Figure below show the Fourier transform of y[n]

65

Interpolation Filter Interpolation Filter

Specifications Specifications

• On the other hand, if we pass x[n]through a factor-of-Lup-sampler generating , the relation between the Fourier transforms of x[n]and are given by

] [n x_u ]

[n x_u

) ( )

( ^j ^j ^L

u e X e

X ^ω = ^ω

1X_u(e^j^ω)

66

Interpolation Filter Interpolation Filter

Specifications Specifications

• It therefore follows that if is passed through an ideal lowpass filterH(z)with a cutoff at π/Land a gain of L, the output of the filter will be precisely y[n]

] [n x_u

) (e^j^ω LH

(12)

67

Interpolation Filter Interpolation Filter

Specifications Specifications

• In practice, a transition band is provided to ensure the realizability and stability of the lowpass interpolation filterH(z)

• Hence, the desired lowpass filter should have a stopband edge at and a passband edge close to to reduce the distortion of the spectrum ofx[n]

s =π/L ω

ωs

ωp

68

Interpolation Filter Interpolation Filter

Specifications Specifications

• If is the highest frequency that needs to be preserved in x[n], then

• Summarizing the specifications of the lowpass interpolation filter are thus given by

ωc

c L

p=ω / ω

⎩⎨

⎧ π ≤ω≤π ω

≤

= ω

ω

L L e L

H ^j ^c

/ ,

/ ) ,

( 0

69

Decimation Filter Decimation Filter

Specifications Specifications

• In a similar manner, we can develop the specifications for the lowpass decimation filter that are given by

⎩⎨

⎧ πω≤ω≤ω≤π

ω =

M e M

H ^j ^c

/ ,

/ ) ,

( 0

1

70

Filter Design Methods Filter Design Methods

• The design of the filterH(z) is a standard IIR or FIR lowpass filter design problem

• Any one of the techniques outlined in Chapter7 can be applied for the design of these lowpass filters

Filters for Fractional Sampling Filters for Fractional Sampling

Rate Alteration Rate Alteration

• For the fractional sampling rate structure shown below, the lowpass filterH(z) has a stopband edge frequency given by

L H(z) M

⎟⎠

⎜ ⎞

⎝

= ⎛

M

s L

π ω min π,

Computational Requirements Computational Requirements

• The lowpass decimation or interpolation filter can be designed either as an FIR or an IIR digital filter

• In the case of single-rate digital signal processing, IIR digital filters are, in general, computationally more efficient than equivalent FIR digital filters, and are therefore preferred where computational cost needs to be minimized

(13)

73

Computational Requirements Computational Requirements

• This issue is not quite the same in the case of multirate digital signal processing

• To illustrate this point further, consider the factor-of-Mdecimator shown below

• If the decimation filter H(z)is an FIR filter of length Nimplemented in a direct form, then

] M [n

x H(z) v[n] y[n]

∑

⁻

=

−

= ¹

0 N m

m n x m h n

v[ ] [ ] [ ]

74

Computational Requirements Computational Requirements

• Now, the down-sampler keeps only every M-th sample of v[n]at its output

• Hence, it is sufficient to compute v[n]only for values of nthat are multiples of Mand skip the computations of in-between samples

• This leads to a factor of Msavings in the computational complexity

75

Computational Requirements Computational Requirements

• Now assume H(z)to be an IIR filter of order Kwith a transfer function

where

) (

) ) ( ) ( (

) (

z D

z z P z H X

z

V = =

K n n

nz p z

P ⁻

∑

=

0

) (

K n n

nz d z

D ⁻

∑

=

+

=

1

) 1 (

76

Computational Requirements Computational Requirements

• Its direct form implementation is given by

• Since v[n]is being down-sampled, it is sufficient to compute v[n]only for values of nthat are integer multiples ofM

−L

−

= [ ] [ ]

]

[n d₁wn 1 d₂wn 2 w

] [ ]

[n K xn

w

d_K − +

−

] [ ]

[ ] [ ]

[n p wn pwn p wn K

v = ₀ + ₁ −1 +L+ _K −

77

Computational Requirements Computational Requirements

• However, the intermediate signal w[n] must be computed for all values ofn

• For example, in the computation of

K+1 successive values ofw[n] are still required

• As a result, the savings in the computation in this case is going to be less than a factor ofM

] [ ]

[ ] [ ]

[M p wM pwM p wM K

v = ₀ + ₁ −1 +L+ _K −

78

Computational Requirements Computational Requirements

• Example-We compare the computational complexity of various implementations of a factor-of-Mdecimator

• Let the sampling frequency be

• Then the number of multiplications per second, to be denoted as , are as follows for various computational schemes

FT

RM