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Introduction to Multirate Systems 1

Chapter I

Introduction to

Multirate Systems

Gordana Jovanovic-Dolecek

INAOE, Mexico

Copyright 2002, Idea Group Publishing.

This chapter treats the fundamentals of multirate system theory and

includes decimation and interpolation as the basic concepts behind the

changing of the sampling rate. The conversion of the sampling rate by an

integer as well as by a rational number is explained. Also discussed are some

widely used interconnections of upsamplers and downsamplers.

INTRODUCTIONThe process of converting the given rate of a signal into a different rate

is calledsampling rate conversion. Systems that employ multiple sampling

rates in the processing of digital signals are called multirate digital signal

processing systems.

Sample rate conversion is one of the main operations in a multirate

system. This chapter focuses on the description of sampling rate conversion

in both time and frequency domains.

DECIMATIONThe reduction of a sampling rate is called decimation, because the

original sample set is reduced (decimated).

Decimation consists of two stages: filtering and downsampling, as

shown in Figure 1.

We will first consider downsampling and through its description illus-trate why it is necessary to apply filtering prior to downsampling.

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2 Jovanovic-Dolecek

DownsamplingDownsamplingreduces the input sampling ratef

iby an integer factorM,

which is known as a downsampling factor. It is customary to use a box with

a down-pointing arrow, followed by the downsampling factor, as a symbol to

represent downsampling, as shown in Figure 2.

The output signaly(m) is called a downsampled signaland is obtained by

taking only every M-th sample of the input signal and discarding all others,

)()( mMxmy = . (1)

To model the downsampling process, it is convenient to divide it into twosteps. The output of thefirststep is the signalx(n), which is obtained by

setting all samples whose indices are not integer multiples ofMto zero. In the

secondstep, all zeros that were introduced in the preceding step are now

discarded, and the downsampled signal is obtained.

The downsampling operation is not invertible because it requires setting

some of the samples to zero. In other words, we cannot recoverx(n) fromy(m)

exactly, but can only compute an approximate value.

Step One

We observe that the sampling rate is not altered during thefirststep, so

that the signalsx(n) andx(n) have the same sampling rate. The signalx(n)

can be considered as a multiplication ofx(n) with the discrete sampling

functioncM(n), where Mdenotes the downsampling factor

)()()(' ncnxnx M= , (2)

where,

,...1,0,1...,;0

1

)( =

=

= motherwise

mMn

ncM . (3)

Mx(n)u(n)

Filtering Downsampling

y(m)

DECIMATION

h(n)

Figure 1: Decimation

Mx(n) y(m)=x(mM)

fi f =f /M0 i

Figure 2: Downsampling

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The sampling function cM(n) is periodic with period M, and as such can be

represented by the Fourier series expansion (see Oppenheim and Schafer,

1989, pp.515-516),

MknjM

k

M ekCM

nc21

0

)(1

)(

=

= (4)

where C(k) are complex-valued Fourier series coefficients defined by

=

=1

0

2

)()(M

n

M

knj

M enckC

. (5)

Substituting (3) into (5) it follows that C(k)=1 for all k. Hence,

=

=1

0

21)(

M

k

Mknj

M eM

nc

. (6)

To analyze the frequency representation of thefirststep of downsampling, let

us now compute the Fourier transform (FT) of the sequencex(n). Using (2)

we have

nj

n

Mnj

n

j encnxenxeX

=

= == )()()(')('

. (7)

Using the relationship established in (6), the above becomes

=

=

=n

M

k

njM

knj

j eeM

nxeX1

0

2

)1

)(()('

. (8)

Finally, by interchanging the sums in (8), the following expression for X(ejw)

results in

=

=

=

1

0

)2

(

)(1

)('M

k n

M

kjn

jenx

MeX

. (9)

Let us now compute the Fourier transform ofx(n),

=

=n

jnj enxeX )()( . (10)

Applying the frequency-shift property of FT, we have

=

=

n

M

kjn

M

kj

enxeX)

2()

2(

)()(

. (11)

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4 Jovanovic-Dolecek

Noticing that the expression on the right side of (11) looks like the one given

in (9), we can conclude that

=

=

1

0

)2

(

)(

1

)('

M

k

M

kj

j

eXMeX

. (12)

Observe that the amplitude is scaled by 1/Mand that the replicas of the

input spectrum are introduced at multiples of2/M.

Step Two

The zeros previously introduced now have to be eliminated. This

operation does not change the content of the signal x(n), rather it just

introduces time scaling by a factor of1/M(samplesx(mM) become y(m)).

Because the operations in time and frequency are inverse to each other,the frequency scale will be multiplied by M, i.e., the spectrum X(ejw/M)

becomes Y(ejw), as shown below.

Using the definition of FT forx(n) we have

.)(')('

=

=n

Mjn

Mj

enxeX

(13)

Becausex(n) is nonzero only forn=mM, we can write

)(')(')(')/(/

=

=

==m

mj

m

mMMjMjeMmxeMmxeX

. (14)

Finally, substituting (1) in (14) and using the definition of FT fory(m) we

arrive at

=

==m

jjmMj eYemyeX ).()()(' / (15)

Using (12) we can rewrite (15) as

=

==

1

0

)2

(

)(1

)(')(M

k

M

kj

jjMeX

MeXeY

. (16)

Sometimes it is more convenient to express the downsampled signal in terms

ofitsz-transform. For a given sequencex(n), itsz-transform is defined as

,)()(

=

=n

nznxzX (17)

wherezis a complex variable, given byjrez= .

-

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It is well known that when it exists, the Fourier transform is simplyX(z)

withjez= ,

)()(

j

ez eXzX j=

= . (18)Consequently, we can express the relationship in (16) in terms of the z-

transforms,

=

=

===1

0

1

0

/2 )(1

)(1

)(')(M

k

M

k

kM

MkjM zWXM

zeXM

zXzY (19)

where

.1,...,0,

2

==

MkeW Mkj

k

M

(20)

Example 1

To illustrate downsampling in the time domain consider the input signal

shown in Figure 3(a) with the downsampling factorM=2. The corresponding

sampling function forn=0,...,16, is given in Figure 3(b). Figure 3(c) demon-

strates thefirststep of downsampling where every second sample is set to

zero. In thesecondstep (Figure 3(d)) we eliminate zeros introduced in the

previous step to finally obtain the downsampled singal. Observe that the

downsampled signal is a compressed-in-time version of the input signal.Let us now consider the frequency domain representation of

downsampling, as shown in Figure 4. During the firststep (Figure 4(b)), one

image (M-1=1) is introduced in the interval [0,2], and the amplitude of the

spectrum is scaled by . In the second step, shown in Figure 4(c), the

frequency is scaled by M=2, so, for example, the frequency points 0.2 and 0.5

become 0.4 and 1, respectively.

We can see that the spectrum of the downsampled signal is the input

spectrum stretched by 2. Remember that in the time domain the opposite was

true (the resulting signal was compressed by a factor of2), because the timeand frequency representations are inverse to each other.

AliasingAs we have already demonstrated, the individual spectra obtained during

the first step of downsampling are the repeated replicas of the original

spectrum. However, if the original signal is not bandlimited to /M, the

replicas will overlap (see Figure 4(b)). This overlapping effect is called

aliasing. In order to avoid aliasing, it is necessary to limit the spectrum of thesignal before downsampling to below /M. This is why a lowpass digital filter

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6 Jovanovic-Dolecek

0 2 4 6 8 10 12 14 16

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

FIRST STEP

n

x'(n)

0 2 4 6 8 10 12 14 160

0.2

0.4

0.6

0.8

1

1.2SAMPLING FUNCTION

n

c2(n)

0 2 4 6 8 10 12 14 16

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

SECOND STEP

n

y(n)

0 2 4 6 8 10 12 14 16

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

INPUT SIGNAL

n

x(n)

Figure 3: Downsampling in time domain

(a) Input signal (b) Discrete sampling function

(c) First step (d) Second step

(from Figure 1) precedes the downsampler. This filter is called a decimation

orantialiasingfilter. The exact filter specifications depend on how much

aliasing, if any, is permitted. Nonetheless, some applications allow a certain

degree of aliasing during downsampling.

The specifications for the lowpass decimation filter are given by (see

Mitra, 2001),

=

M

MeH

cj

/,0

/,1)( , (21)

where c

represents the highest frequency that needs to be preserved.

Properties of DownsamplingLet us first verify that the downsampling is a linear and time-varying

operation. The operation is linear if the principle of superposition holds. That

means that downsampling of the sum of the scaled input signals must be equalto the scaled sum of the downsampled signals.

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1INPUT SIGNAL

w/pi

abs(X)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5FIRST STEP

w/pi

abs(X')

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5SECOND STEP

w/pi

abs(Y)(a) Input signal

(b) First step

(c) Second step

Figure 4: Downsampling in frequency domain

Structure 1

MX(z) Y(z )

M

X (z)1a1

X (z)NaN

MY (z)1

Y(z )M

X (z)1a1

X (z)NaN

MY (z)N

Structure 2

Figure 5: Superposition principle

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8 Jovanovic-Dolecek

The proof is obvious, so we omit it. Therefore, downsampling is a linear

operation.

Now we examine whether downsampling is a time-varying operation. In

general, if the delayD of the input signal results in the same delayD of theoutput signal, the operation is time invariant, otherwise it rendered time-

varying. According to the equation (1) if the input signal is delayed byD,

),( DmMx (22)and the downsampled signal will be

).()()( DmyM

Dmy

M

DmMy =

(23)

Consequently, the delay in the input signal does not result in the same

delay of the downsampled signal, and we can conclude that downsampling isa time-varying operation. When D/Mis not an integer, the downsampled

signal will have a fractional delay, as (23) shows. This means that the

downsampled signal with no input delay and the downsampled delayed signal

will have a different shape.

Consider the case when the delay is a multiple of the downsampling

factorM, i.e., whenD=kM. It easily follows from (23) that the downsampled

signal will be

).()( kmyM

kMmy = (24)

In this case, the downsampling of the input signal that is delayed by kM,

results in a delayed-by-k version of the original downsampled signal. As we

have already seen, the delay is not the same in the input and the output.

Nonetheless, at the output we do obtain a delayed downsampled version of the

input signal. This means that if a delay is a multiple of the downsampling

factor M, the corresponding downsampled signal will have the same shape as

a downsampled signal with no input delay.

Useful IdentitiesThree useful identities summarize the important properties associated

with downsampling. The first identity states that the sum of the scaled,

individually downsampled signals is the same as the downsampled sum of

these signals. This property follows directly from the principle of superposi-

tion, as previously discussed.

The second identity (Figure 6) establishes that a delay ofMsamples

before the downsampler is equivalent to a delay of one sample after the

downsampler (where Mrepresents the downsampling factor).

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Structure 1 Structure 2

MX(z) Y(z )

M

z-M

MX(z) Y(z )

M

z-1

Y (z )1M

Figure 6: Second identity

Consider Structure 1 in Figure 6. We see that the input to the downsampler is

X(z) z-M, so that according to (19), the output of Structure 1 is given by

.))((1

)(1

0

MM

k

kM

kM

M zWzWXM

zY

== (25)

Using (20) it follows that

,1=

kM

MW (26)which simplifies the above expression into

.)(1

)(1

0

=

=M

k

kM

MM zWXzM

zY (27)

We now consider Structure 2 in Figure 6. The output of the downsampler is

(note thatzbecomeszMafter downsampling),

.)(1

)(1

0

1

=

=M

k

kM

MzWX

M

zY (28)

Consequently, the output of this structure is

.)(1

)()()(1

0

11

=

==M

k

kM

MMMM zWXM

zzYzzY (29)

Comparing the expressions in (27) and (29), it follows that Structures 1 and

2 in Figure 6 are equivalent.

Figure 7 demonstrates the third identity. The filterG(zM) is called an

expanded filterand is obtained by replacing each delay z-1 of the original filter

G(z) with a delay z-M.In time domain this is equivalent to inserting M-1 zeros

between the original samples of the impulse response.

The identitystates that the filtering by the expanded filter followed by

downsampling is equivalent to having downsampling first, followed by the

filtering with the original filter. To confirm this algebraically, consider the

structures shown in Figure 7.

Using (19) and (26) we arrive at the output of the first structure

.)()(1

)()(1

)(

1

)(

1

0

1

0

1

01

=

=

=

==

=M

k

MkM

kMM

MM

k

kM

M

k

k

M

M

zGzWXM

WzGzWXM

zWXMzY(30)

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10 Jovanovic-Dolecek

Similarly, using (19), the output of the downsampler in the second

structure is

=

=1

0

1 )(1

)(M

k

kM

M zWXM

zY . (31)

Finally, the output of the second structure is

=

== 1

0

1 )()(1)()()(

M

k

MkM

MMM zGzWXM

zGzYzY . (32)

Comparing (30) and (32), we can conclude that the two structures in Figure

7 are equivalent.

Polyphase DecimationWe have previously seen that in the decimation structure the filtering

is performed at a higher rate. If we use an IIR filter for antialiasing, we

would have to compute all of the feedback values for each new input value(Grover and Deller, 1999). On the other hand, FIR filters have no

feedback, so that during the convolution many terms that are not used in

the output signal are needlessly computed. Hence, a more efficient

structure would be if the FIR filtering were performed at a lower sampling

rate. To do this it is necessary to analyze the FIR filter through its

polyphase components.

The Polyphase FiltersIf the FIR filter hasNcoefficients, whereNis an integer multiple of

M, we can obtain M different discretely sampled components of the

impulse response. That means that the correspondingz-transform can be

partitioned into Msubsignals. For example, forN=12 and M=4 we can

write

Structure 2

MX(z) Y(z )

M

G( )z

Structure 1

MX(z) Y(z )

M

G(z )MX (z)1 Y (z )1

M

Figure 7: Third identity

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].)11()7()3([z

])10()6()2([

])9()5()1([

])8()4()0([

)11()7()3(

)10()6()2(

)9()5()1(

)8()4()0()()(

8403-

8402

8401

8400

1173

1062

951

8401

0

=

+++

+++

+++

++=

+++

++++++

++==

zhzhzh

zhzhzhz

zhzhzhz

zhzhzhz

zhzhzh

zhzhzh

zhzhzh

zhzhzhznhzHN

n

n

(33)

This equation can now be rewritten to yield

=

=3

0

4),()(

k

kk zHzzH (34)

where

.3210;))(4()(2

0

44 ,,,kzknhzHn

nk =+=

=

(35)

We can generalize the expression (34) to obtain

=

=1

0

),()(M

k

Mk

kzHzzH (36)

where

=

=+=1/

0

1,...,0;))(()(MN

n

nMMk MkzknMhzH . (37)

A realization ofH(z) based on the decomposition of (36) is called a

polyphase realization and designates the basis for the polyphase decimation.

The equation (37) represents thepolyphase components of the filter. From the

expressions in (33) and (37) we can notice that the polyphase components are

obtained by the left shift of the impulse response by k samples, followed by

discrete sampling. The zero component corresponds to the shift k=0, the first

one to the k=1, and the M-1-th to k=M-1.

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By eliminating zeros introduced by discrete sampling, from (37) (recall

that this is the second step of downsampling), we arrive at

=

=+=1/

0 1,...,0;)()(

MN

n

n

k MkzknMhzH (38)

Notice that the polyphase components in (37) are simply the expanded

polyphase components given by (38). Therefore, the representation in (38)

can be used in the applications of the third identity.

Utilizing the relation (36), the decimation structure from Figure 1 can be

redrawn as in Figure 8(a). Using thefirstand third identities we obtain a more

efficient version in which both the number of filter operations as well as the

amount of memory required are reduced by a factor ofM. This structure is

shown in Figure 8(b). As we can see, the subfilters are polyphase components(38) of the decimation filter. Also notice that the filtering is now performed

at the lower sampling rate.

(a)

MU(z) Y(z)

H (z )0M

H (z )1M

H (z )2M

H (z )M-1M

...

...

...

H(z)

z-1

z-1

Figure 8: Polyphase decimation

MU(z) Y(z)

H (z)0

H (z)1

H (z)2

H (z )M-1

..

.

..

.

...

z-1

z-1

M

M

M

(b)

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Note that the delay/downsampler chain has the effect of distributing

consecutive samples among the polyphase filters. Due to this, the structure is

often represented using an input commutator as shown in Figure 9.

INTERPOLATIONThe procedure of increasing the sampling rate is called interpolation,

and it consists of two stages: upsampling and filtering (shown in Figure

10). We are going to consider interpolation in detail in both time and

frequency domains.

UpsamplingUpsampling increases the sampling rate by an integer factorL, by

insertingL-1 equally spaced zeros between each pair of samples,

=

=otherwise

mLnforLnxny

0

)/()( (39)

whereL is called an interpolation factor.

The symbol for this operation is a box with an upward-pointing arrow,

followed by the interpolation factor, as Figure 11 illustrates. As we can see,

the input sampling ratefiis increasedL times.

We can observe the following:

1. The process of upsampling does not change the content of the input signal,

and it only introduces the scaling of the time axis by a factorL. (The time

U(z)

Y(z)H (z )0

H (z )1

H (z )2

H (z)M-1

..

.

..

.

0 1

2

M-1

Figure 9: Polyphase decimation with an input commutator

Interpolation

Lx(m) u(n)

h(n)y(n)

Upsampling Filtering

Figure 10: Interpolation

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axis is divided byL). In the frequency domain, according to the principle

of duality, the frequency scale is multiplied byL.

To verify this observation, we will analyze the behavior of the Fourier

transform of aL-fold upsampler.

2. Consequently, the operation of upsampling (unlike downsampling) is

invertible, in other words, it is possible to recover the input signalx(m)

from samples ofy(n) exactly.

Using (39), the Fourier transform of y(n) is given by

)()/(

)/()()(

)/)((

/

Lj

n

LnLj

LnLj

n n

njj

eXeLnx

eLnxenyeY

==

==

=

=

=

(40)

We can notice that only frequency scaling, but no amplitude scaling is

introduced. Why is it the case that in upsampling, unlike in downsampling,

there is no change in the amplitude scale?In upsampling we simply add zeros between samples, and thus the

sample amplitude of the signal remains unchanged. (Remember that in

downsampling we have set some of the samples of the signal to zero.) On the

other hand, the insertion of zeros results in a new time scaling, and conse-

quently, in a new frequency scaling. This means that a given frequency inX(ej) is transformed into a new frequency /L in Y(ej). As a result of thisoperation,L-1 unwanted images of the input signal spectra are introduced in

the interval [0, 2].

It is often instructive to describe the process of upsampling in terms of the z-transform. Using (18) and (40) we obtain z-transform of the upsampled signal

)()( LzXzY = . (41)

Example 2

We consider a signal shown in Figure 12(a). By insertingL-1=2 zeros

between each pair of samples, the sampling rate is increased by L=3. The

result of this operation is shown in Figure 12(b).

Figure 13(a) illustrates the spectrum of the input signal, while Figure 13(b)shows the spectra of the upsampled signal. We can notice that the spectrum

Lx(m) y(n)

fi f L0=fi

Figure 11: Upsampling

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Figure 12: Upsampling in time domain

0 2 4 6 8 10 12 14 16

0

0.05

0.1

0.15

0.2

0.25

0.3

m

x(m)

INPUT SIGNAL

0 5 10 15 20 25 30 35 40 45

0

0.05

0.1

0.15

0.2

0.25

0.3

n

y(n)

UPSAMPLED SIGNAL

(a) Input Signal (b) Upsampled signal, L = 3

Figure 13: Upsampling in frequency domain

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1INPUT SIGNAL

w/pi

abs(X)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

w/pi

abs(Y)

UPSAMPLED SIGNAL

(a) Input signal (b) Upsampled signal, L = 3

has no change in amplitude scale and thatL-1=2 images are introduced in the

interval [ 0,2].

ImagingThe process of upsampling introduces the replicas of the main spectra at

every 2/L. This is called imaging, since there are L-1 replicas (images) in 2

(see Figure 13(b)). In order to remove the unwanted image spectra we need

a lowpass filter immediately after upsampling. This filter is called an anti-

imaging filter. In the time domain, the effect is that the zero-valued samples

introduced by upsampling are filled with interpolated values. Because of

this property, the filter is also called an interpolation filter.

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The specifications for the interpolation filter are given by Mitra (2001)

=

L

LLeH

cj

/,0

/,)( , (42)

wherecis the highest frequency that needs to be preserved in the interpolated

signal. Similarly to an anti-aliasing filter, an anti-imaging filter is usually

designed as an FIR filter (Mitra, 2001).

Comparing (21) and (42) we can notice that there is no duality between

the gain of the anti-aliasing filter and the gain of the anti-imaging filter. Why

does an anti-imaging filter have a gain L? The explanation is given in the

following paragraphs.

Assume that x(m) (see Figure 10) has been obtained by sampling abandlimited continuous signalx

c(t) at the Nyquist rate. The Fourier transform

of the discrete signal X(ej) is related to the Fourier transform of the

continuous signalXc(j), ( is a continuous frequency), as

xxk x

c

x

j

TT

kj

TjX

TeX

==

=

);2

(1

)( , (43)

where Tx

denotes the sampling period.

In a similar way, we can assume that the interpolated signal u(n) in Figure

10 is also obtained by sampling the same continuous signal with a samplingperiod, T

y=T

x/L. Its Fourier transform is related toX

c(j) as

yxk x

c

xyk y

c

y

j

TT

kL

T

LjX

T

L

T

k

T

jX

TeU

===

=

=

);2

()2

(1

)( . (44)

The output signal u(n) has been passed through a low pass filter, and so

only the base spectrum is retained. Therefore, in (44) all spectral components

other than k=0, are eliminated, which gives

)()(x

c

x

j

T

LjXT

LeU

= , (45)

or using (43)

)()( Ljj eLXeU = . (46)Because no amplitude scaling was introduced during upsampling, the gainL

in (46) must be compensated for during the filtering stage.

Properties of Upsampling

In this section we will verify that upsampling is a linear and time-varyingoperation. The linearity property is illustrated in Figure 14. Since it is

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straightforward to verify, we will state without a proof that upsampling is a

linear operation.

We will now consider whether this operation is time-varying. Let us

suppose that the input of the upsampler (Figure 11) has a delay ofD samples,)( Dmx . (47)

The upsampled signal will be

)()()())(( DnyDLnyDLmLyLDmy == . (48)Consequently,upsampling is a time-dependent operation. Since the delay DL

is an integer, the upsampled delayed signal and the upsampled signal with no

delay will always have the same shape. Remember that in the case of a

downsampled signal, this relationship holds only when the delay is a multiple

of M.

Useful IdentitiesWe have already seen three useful identities of the downsampled signals,

and now we will state the corresponding identities associated with upsampling.

Thefourth identity asserts that the output signals Y1(z),...,Y

N(z), which are

obtained by upsampling the input signal and then scaling by a1,...,a

N, respec-

tively (the first structure in Figure 15) willgive the sameresult as if the signal

is first scaled and then downsampled (see the second structure in Figure 15).

The proof is obvious so we omit it.Thefifth identity states that a delay of one sample before upsampling is

equivalent to the delay ofL samples after upsampling (Figure 16).

In the first structure we have:

11 ))(()(

= LLL zzXzX . (49)The output of the first structure yields

LLL zzXzXzY == )()()( 1 . (50)

At the upsampler output in the second structure it holds that

)()(1LzXzY = , (51)

Structure 1

LX(z) Y(z)

X (z)1a1

X (z)NaN

Y (z)1

Y(z)

X (z)1a1

X (z)NaNY (z)N

Structure 2

L

L

Figure 14: Superposition principle

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resulting in

LLL zzXzzYzY == )()()( 1 , (52)at the output. Since the above expression equals the one given by (50), the

equivalency of the two structures is confirmed.

Thesixth identity, which is a more general version of thefifth identity,

states that filtering followed by upsampling is equivalent to having upsamplingfirst followed by expanded filtering. It is illustrated in Figure 17.

In the first structure (Figure 17)we have

)()()()(

)()()(

1

1

LLLzGzXzXzY

zGzXzX

==

=(53)

Similarly, from the second structure we obtain

)()()()()(

)()(

1

1

LLL

L

zGzXzGzYzY

zXzY

==

=(54)

Since the equations (53) and (54) are identical, the two structures are equivalent.

Polyphase InterpolationWe notice that in the interpolation structure, the filtering is performed

at the higher sampling rate. That means that most of the data values going

into the filter are zero, more precisely L-1 out of every L value. As a

consequence, in the process of convolution there are many unnecessarymultiplications with zero.

Structure 1

LX(z)

Y (z)1a1

Y (z)NaN

Y (z)1

X(z)

a1

aNY (z)N

Structure 2

L

L

Figure 15: Fourth identity

Structure 1

X(z )L

Y(z)z

-1L

X (z )1L

Structure 2

X(z )L Y(z)

z-LL

Y (z)1

Figure 16: Fifth identity

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The convolution at the higher sampling rate can be replaced by indepen-

dent convolutions at the lower input sampling rate using polyphase decompo-

sition (36). We can simply replace the coefficient MbyL to obtain

=

=1

0

)()(L

k

Lk

kzHzzH . (55)

If we express the polyphase filter in the interpolation structure in Figure10, using the above expression, we obtain the first structure in Figure 18 (a).

Its transpose (see Mitra, 2001, p.352), is shown in Structure 2 in Figure 18(a).

According to thefourth identity, we can place the upsampler into parallel

branches, and then using thesixth identity we can interchange filtering and

upsampling, as illustrated in Figure 18(b). Each input sample is simulta-

neously sent to the subfilters Ho(z),....H

L-1(z), where filtering is performed at

the lower sampling rate giving the components Xo(z), ..., X

L-1(z). These

components are upsampled (filled with zeros), and then interleaved at the

higher sampling rate using a series of delays, and finally are combined to yieldthe output signal.

Some authors also use a commutator in the output to present the

combining of the componentsXo(z), ..., X

L-1(z) into an output signal. Figure 19

shows a polyphase interpolator with an output commutator.

INTERCONNECTION OF

SAMPLING RATE CONVERTERSIn this section we will discuss some of the widely used interconnec-

tions of upsamplers and downsamplers in the multirate systems. An

interchange of sampling converters can often lead to a computationally

more efficient realization.

Cascade of Sampling Rate ConvertersFirst we consider the cascade of an L-fold upsampler and an L-fold

downsampler, as shown in Figure 20(a).

It is straightforward to conclude that the output is equal to the input.

X(z )

L

Structure 1

Y(z)G(z) L

X (z )1L

Structure 2

X(z) Y(z)G( )z

LL

Y1(z)

Figure 17: Sixth identity

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LX(z )

L

Y(z)H (z )0

L

H (z )1L

H (z )2L

H (z )L-1L

...

...

...

z-1

z

-1

Structure 1

H(z)

...

z-1

z-1

LX(z )

L

Y(z)

...

H (z )0L

H (z )1L

H (z )2L

H (z )L-1L

...

Structure 2

H(z)

Figure 18: Polyphase interpolation

x(m)y(n)

H (z)0

H (z)1

H (z)2

H (z )L-1

..

.

..

.

x (m)0

x (m)1

x (m)2

x (m)L-1

k=0

k=L-1

1

2

Figure 19: Polyphase interpolator with an output commutator

Upsampling by a factorL introducesL-1 zeros after each sample of theinput signal, and downsampling by a same factorL removes those zeros,

and the net result is the original signal. If upsampling precedes

downsampling, where both operations have the same factor, namely when

M=L, the signal is not changed.

Now we look at the cascade in Figure 20(b). Does the output signal still

equal the input signal? In the firststep of downsampling we set all samples

whose indices are not integer multiples ofMto zero, and, then, in thesecond

step, we remove those zeros. Upsampling by Magain introduces those zeros,

and therefore this result corresponds to the firststep of downsampling. If

LX(z) Y(z)

H(z)0

H(z)1

H(z)2

H (z )L-1

.

.

.

.

.

.

.

.

.

z-1

z-1

L

L

L

X (z)0

X (z)1

X (z)L-1

(b)

(a)

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Figure 20: Cascading sampling converters

L LX(z )

LX (z)1 Y(z )

L

(a) Upsampler-downsampler

(b) Downsampler-upsampler

MMX(z)X(z) X (z )1

MY(z)

downsampling is performed before upsampling, with the same sampling

factor, the output signal will be different from the input signal, and the

operation will be equivalent to discrete sampling.

Interchanging of the Sampling Rate ConvertersWe are going to consider whether the operations of downsampling and

upsampling can be interchanged, i.e., whether they are commutative or not.

Structure (a) in Figure 21 shows the operation of downsampling by a factor

ofM, followed by upsampling by a factor ofL. The question is when we can

perform upsampling by L first, followed by downsampling by M, (Figure 21

(b)) and still obtain the same result. The condition for the equivalency of the

structure (a) and (b) will provide the answer.

Structure (a):

.)(1

)()(

)(1

)(;)(1

)(

1

0

/11

1

0

/11

1

0

1

=

=

=

==

==

M

k

kM

MLL

M

k

kM

MM

k

kM

M

WzXM

zXzY

WzXM

zXzWXM

zX

. (56)

Figure 21: Interchanging of sampling rate converters

M LX(z)X(z) X (z)1 Y (z)1

(a) Downsampler-upsampler

(b) Upsampler-downsampler

L MX(z)X(z) X (z)2 Y (z)2

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Structure (b):

.)(1

)(

)(1)(1)(

)()(

1

0

/2

1

0

1

0

22

2

=

=

=

=

==

=

M

k

kLM

ML

M

k

kLM

LM

k

kM

M

L

WzXM

zY

WzXM

zWXM

zY

zXzX

. (57)

Obviously,M-fold downsampler can be interchanged with anL-fold upsampler

if and only ifY1(z)=Y

2(z). From (56) and (57) it is clear that this will only occur

when

kLM

kM WW = . (58)

This relation holds if and only ifMandL are relatively prime, i.e., Mand

L do not have any common integer factor except for1. For the proof, see Suter

(1998).

Example 3

Let us illustrate the former result by lettingL=2 and M=3, so that the

sampling factors are relatively prime.

{ }

{ }3/23/83/40322

23

3/43/23

2

3

;;2,1,0;

;;2,1,0;

jjjj

kj

k

jjjo

kj

k

eeeekeW

eeekeW

==

==

=

==

(59)

As expected, two sets in (59) are equivalent. Now we suppose thatL=2 andM=4,

i.e., MandL are not relatively prime since M=2L. The corresponding sets are

{ }

{ } { }

32024

2/32/024

;;;3,2,1,0;

;;;3,2,1,0;

jjjjkjk

jjjj

kj

k

eeeekeW

eeeekeW

===

=

==(60)

L LH(z) X(z) X (z)1 Y (z)1 Y(z)

LTI

Structure 1

H (z)0 X(z) Y(z)

Structure 2

Figure 22: Cascade Upsampler-LTI-Downsampler

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Unsurprisingly, two sets in (61) are not equivalent.

In summary, using (56)-(58), M-fold downsampler and L-fold upsampler

can be interchanged if and only if M and L are relatively prime.

Sampling Rate Converters and the LTI System

Upsampler-LTI-Downsampler

We will consider an extended version of the previous structure in Figure

20(a), with an LTI system between converters, as shown in Figure 22.

The analysis of the first structure gives

=

=

=

=

=

===

==

=

1

0

/1

1

0

1

0

1

0

1

11

1

)].(1

)[()(

)(1

)()()(1

)(1

)(

)()()()()(

)()(

L

k

kL

L

L

k

kL

LL

k

kL

kLL

LL

k

kL

L

L

L

WzHL

zXzY

zWHL

zXzWHWzXL

zWYL

zY

zHzXzHzXzY

zXzX

.

(61)

The factor inside the middle parenthesis in the last equation (61) is a

downsampled filter impulse response (see equation (19)), and thus, accordingto (38) is equal to the 0-th polyphase component.

=

=1

0

/10 )(

1)(

L

k

kL

LWzH

LzH . (62)

Using (62) we finally arrive at

)()()( 0 zHzXzY = . (63)According to the last relation, the cascade in the first structure is

equivalent to the second structure, where the equivalent filtering uses only the

0-th polyphase component (38) of the filterH(z) (the remaining polyphase

components ofH(z) are not used). For this reason, the structure is also called

thepolyphase identity (Vaidyanathan, 1993). The result is an engaging one

because the upsampler and downsampler are individually time variant sys-

MM H(z) X(z) X (z)1 Y (z)1 Y(z)

LTI

H(z )M

Y(z)M

X(z)

Structure 1 Structure 2

MX (z)

1

Figure 23: Cascade downsampler-LTI-Upsampler

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tems, yet the resulting structure is time invariant.

Downsampler-LTI-Upsampler

Let us now analyze the extended version of the structure 20(b), contain-ing an LTI system between sampling converters, as shown in Figure 23.

Applying the sixth identity, we see that the final output results from

filtering the output of the cascade downsampler/upsampler (discrete sam-

pling) with an expanded filterH(zM), as shown in the equivalent Structure 2

in Figure 23. Therefore, unlike the polyphase identity, this equivalent struc-

ture is now time variant.

Rational Sampling Rate Conversion

Thus far we have discussed changing the sampling rate by integer factors,decimating by M, or interpolating byL. In this section we consider changing

the sampling rate by a ratio of two integers,L/M. Obviously we can perform

this by cascading decimation by a factorMand interpolation by a factorL.

There are two possible cascade connections depending on what is

performed first, decimation or interpolation, as shown in Figures 24(a) and

(b). In the structure where the interpolation is performed before decimation

(Figure 24(b)), we can notice that the interpolation and the decimation filters

are both lowpass filters operating at the same sampling rate. Thus, they can

be combined into one filterG(z), as shown in Figure 24(c), to yield the desired

structure for the rational sampling rate conversion. The result is shown in

Figure 24 (c).

Figure 24: Rational sampling conversion

M LH (z)1 H (z)2X(z) Y(z)

DECIMATION INTERPOLATION

L MH (z)1 H (z)2X(z) Y(z)

L MG(z)X(z) Y(z)

(a)

(b)

(c)

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Lowpass filterG(z) has a normalized stopband cutoff frequency at

)/,/( MLmins = , (64)which suppresses the imaging caused by upsampling and at the same time

eliminates the aliasing caused by downsampling.Recently, a new efficient time-varying filter structure for rational

sampling rate conversion was derived in Webb (2000). This new structure

uses a transposed filter structure, and is particularly suitable for DSP

implementations.

An efficient algorithm for the sampling rate conversion from 44.1kHz

compact disc (CD) to 48 kHz digital audio tape (DAT) has been proposed by

Rajamani et al. (2000). The method uses digital interpolation followed by a

fractional delay filter. Another interesting result can be found in a paper by

Russel (2000) where a new rational sampling rate conversion using IIR filters

is presented.

SUMMARYReduction of the sampling rate is called decimation, and it consists of two

stages: filtering and downsampling. Downsampling reduces the input sam-

pling rate by an integer factorM, and is a linear and time variant operation. A

downsampled signal is a shortened time, frequency stretched version of theinput signal. To avoid the aliasing, the filtering is introduced to bandlimit the

input signal. Polyphase decimation, which utilizes polyphase components of

a decimation filter, is a preferred structure for decimation because it enables

filtering to be performed at a lower sampling rate.

The procedure of increasing the sampling rate is called interpolation, and

it has two parts, upsampling and filtering. Upsampling increases the input

sampling rate by an integer factorL, by insertingL-1 equally spaced zeros

between each pair of input samples. This operation is a linear and time variant

operation and, unlike downsampling, is invertible. This process introducesthe replicas of the main spectra at every 2/L, which is called imaging. In order

to remove these images, a lowpass filter must be placed immediately after

upsampling. A more efficient interpolation structure is obtained by using the

polyphase components of the interpolation filter, so that the filtering is

performed at lower sampling rate.

If upsampling precedes downsampling, where both operations have the

same factor, namely M=L, the input signal is not changed. However, if

downsampling is performed before upsampling, with the same factorL=M,

the equivalent operation corresponds to thefirststep of downsampling, and

thus the output signal is different from the input signal.

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M-fold downsampler andL-fold downsampler can be interchanged if and

only ifMandL are relatively prime.

The cascade Upsampler-LTI systemDownsampler is equivalent to

filtering with the 0-th polyphase component of the LTI, and is called thepolyphase identity. Even though both the upsampler and downsampler are

time variant, the resulting structure is time invariant.

On the other hand, the cascade Downsampler-LTI-Upsampler is time

variant, and is equivalent to discrete sampling with an expanded filter.

Rational sampling conversion can be efficiently performed as a cascade

of upsampling and downsampling, where the interpolation and decimation

filters are combined into one filter.

MATLAB programs which illustrate Chapter 1 are available from the

Web site http://www.inaoep/electronica.

REFERENCESBurrus C. S., McClellan J. H., Oppenheim, A. O., Parks, T. W, Schafer, R. W. and Schuessler

H. W. (1994). Computer-Based Exercises for Signal Processing Using MATLAB.New Jersey: Prentice Hall, Inc.

Crochiere R. E. and Rabiner L. R. (1983). Multirate Digital Signal Processing. New Jersey:Prentice Hall, Inc.

Dabrowski, A. (1997). Multirate and Multiphase Switched-Capacitor Circuits. London:

Chapman & Hall.Fliedge, N. J. (1995). Multirate Digital Signal Processing. Chichester: The McGraw-Hill

Companies, Inc.Grover, D. and Deller, J. R. (1999).Digital Signal Processing and the Microcontroller. New

Jersey: Prentice Hall, Inc.Haddad, R. A. and Parson T. W. (1991).Digital Signal Processing: Theory, Applications,

and Hardware. New York: W. H. Freeman and Company.Mitra, S. K. (2001).Digital Signal Processing: A Computer-Based Approach, (Second edition).

New York: The McGraw-Hill Companies, Inc.Oppenheim, A. V. and Schafer, R. W. (1989). Discrete-Time Signal Processing. New

Jersey: Prentice-Hall, Inc.

Orfanidis, S. J. (1996). Signal Processing. New Jersey: Prentice Hall, Inc.Rajamani K., Lai Y. S. and Farrow C. W. (2000). Efficient algorithm for sample rateconversion from CD to DAT.IEEE Signal Processing Letters, October 10, 7, 288-290.

Russel A, I. (2000). Efficient rational sampling rate alteration using IIR filters.IEEE SignalProcessing Letters, January 1, 7, 6-7.

Stearns, S. D. and David, R. A. (1996). Signal Processing Algorithms in MATLAB. NewJersey: Prentice Hall, Inc.

Suter, B. W. (1998). Multirate and Wavelet Signal Processing. San Diego: Academic Press.Vaidyanathan, P. P. (1993). Multirate Systems and Filter Banks. New Jersey: Prentice Hall,

Inc.Webb, J. L. H. (2000). Transposed FIR filter structure with time-varying coefficients for

digital data resampling.IEEE Transactions on Signal Processing, September 9, 48,2594-2600.

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