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A Multirate Repetitive Control for Fractional-order Servos in

Laser-based Additive Manufacturing

Dan Wang and Xu Chen

Abstract— This paper studies repetitive control (RC)

algorithms to advance the quality of repetitive energy

deposition in laser-based additive manufacturing (AM). An

intrinsic limitation appears in discrete-time RC when the

period of the exogenous signal is not an integer multiple

of the sampling time. Such a challenge hampers high-

performance applications of RC to laser-based AM because

periodicity of the exogenous signal has no guarantees to

comply with the sampling rate of molten-pool sensors. This

paper investigates three RC algorithms to address such

fractional-order RC cases. A wide-band RC and a quasi

RC apply the nearest integer approximation of the period,

yielding overdetermined and partial attenuation of the

periodic disturbance. A new multirate RC generates high-

gain control signals exactly at the fundamental frequency

and its harmonics. Experimentation on a dual-axis galvo

scanner in laser-based AM compares the eﬀectiveness of

diﬀerent algorithms and reveals fundamental beneﬁts of

the proposed multirate RC.

Index Terms— repetitive control, disturbance observer,

additive manufacturing, sampled-data control

I. Introduction

Repetitive control (RC) [1] is designed to track/reject

periodic exogenous signals in applications with repet-

itive tasks. By learning from memories of previous

iterations in the task, RC can drastically enhance cur-

rent control performance in the structured task space

and has been adopted in many ﬁelds, such as tracking

controls in magnetic and optical disk drives [2], wafer

scanners [3], and robotic manipulators [4], as well as

regulation controls in wind turbines [5], power con-

verters [6], and unmanned aerial vehicles [7].

This paper studies RC in laser-based additive manu-

facturing (AM) processes, with a focus on the selective

laser sintering (SLS) subcategory. This AM technology

applies laser beams as the energy source to melt and

join powder materials layer by layer [8]. A typical

workpiece is built from many thousands of thin layers.

Within each layer, laser gets reﬂected by coordinated

mirrors in a beam deﬂection mechanism (e.g., a dual-

axis galvo scanner) to follow trajectories predeﬁned

by the object geometry (in a “slicing” process). This

process contains highly repetitive patterns [9], [10],

with the mirrors driven by periodic or near-periodic

reference signals. As practical servo control systems

inevitably have limited bandwidths, distinct periodic

errors are induced by diﬀerent part geometries and

The authors are with Department of Mechanical Engineer-

ing, University of Connecticut, Storrs, CT, 06269, USA (email:

dan.wang@uconn.edu;xchen@uconn.edu)

laser path plannings. Therefore, RC algorithms, de-

signed to enhance the tracking/regulation control of

periodic references/disturbances, has the potential to

fundamentally improve the quality of the repetitive

scanning of the laser beam. Indeed, such periodicity has

been validated and leveraged upon to improve control

processes in other laser-based AM technologies [11],

[12].

The internal model principle [13] is the theoretical

basis of RC. In digital RC, an internal model 1/1−z−N

is implemented in the controller, where zis the complex

indeterminate in the z-transform and Nis the period

of the disturbance/reference. Nequals the sampling

frequency (denoted in this paper as 1/Tsor fs) divided

by the fundamental frequency ( f0) of the periodic sig-

nal. When Nis an integer, the repetitive controller can

generate high-gain control signals at the fundamental

frequency and its harmonics, yielding small gains in

the error-rejection dynamics. When fsis not divisible by

f0, that is, Nis not an integer, RC with the rounded N

cannot target the aimed harmonic frequencies exactly,

resulting in degraded performance. Unfortunately, such

a fundamental ineﬃciency occurs to SLS AM because

frequency of the exogenous reference or disturbance is

closely related to scanning trajectories and is not guar-

anteed to divide the sampling frequencies of molten-

pool sensors or the galvo scanner.

Several strategies exist to potentially address prob-

lems related to fractional-order periods in RC. High-

order repetitive controllers [14], [15] introduce a series

of delay elements to widen the high-gain region around

the harmonic frequencies. Spatial repetitive controllers

[16], [17] employ a spatial domain to obtain a time-

invariant disturbance period. Adaptive RC schemes

[18], [19] adjust the sampling rate adaptively to get an

integer N. Delay-varying repetitive controllers [20] use

knowledge of the repetitive variable to continuously

adjust the time-varying delay. [6], [21] propose diﬀerent

ﬁlters to approximate the fractional order of delay.

Complementing previous RC designs, this paper

aims at generating enhanced control eﬀorts exactly

at desired frequencies in the fractional-order RC. The

main result is the development of a multirate RC

algorithm along with two indirect RC schemes for

fractional-order periods. First, a wide-band RC is

achieved by applying the nearest integer of Nwhile

widening the attenuation width of each frequency

notch in the error-rejection dynamics. In the second

2018 Annual American Control Conference (ACC)

June 27–29, 2018. Wisconsin Center, Milwaukee, USA

978-1-5386-5428-6/$31.00 ©2018 AACC 4831

indirect quasi RC, the fundamental frequency is shifted

to get an integer N, which creates an overdetermined

rejection of the original repetitive errors. The proposed

new multirate RC designs the internal model under a

second sampling frequency f0

ssuch that N=f0

s/f0is an

integer and hence high-gain control signals are created

at the exact frequencies. The remainder of this paper

will discuss theory, implementation guidance, and per-

formance comparison of the proposed algorithms. In

Section II, we review the basis of plug-in RC design and

introduce cross-channel periodic disturbances in laser-

based AM. The three fractional-order RC algorithms

are built in Section III. Theoretical analyses are veriﬁed

by an experimental study on a dual-axis galvo scanner

platform in SLS. Section IV provides the experimental

veriﬁcation of the algorithms. Finally, Section V con-

cludes the paper.

II. Preliminaries

In this section, a plug-in RC design is ﬁrst reviewed.

Then we introduce the cross-channel periodic distur-

bances in a dual-axis galvo scanner used in SLS.

A. Fundamentals of repetitive control

Consider a discrete-time feedback control system

composed of the plant Pz−1and the baseline con-

troller Cz−1(Fig. 1 with the dotted box removed).

Cz−1can be designed by means of regular servo

algorithms, such as PID, H∞, and lead-lag compensa-

tion. The signals r(k),e(k),d(k), and yd(k)represent

the reference, tracking error, input disturbance, and

system output, respectively. Throughout this paper, we

assume that 1) the coeﬃcients of all transfer functions

are real; 2) both Pz−1and Cz−1are rational, proper,

linear, and time-invariant; 3) the baseline feedback loop

consisting of Pz−1and Cz−1is stable.

C(z−1)P(z−1)

Q(z−1)

z−mˆ

P−1(z−1)z−m

d(k)

+

r(k)+e(k)

+ud(k)+

yd(k)

−

+

w(k)

+

+

Plug-in compensator

Fig. 1. Block diagram for a plug-in RC design.

The proposed RC algorithms are based on a plug-

in RC design [3] shown in Fig. 1. Here, the plug-in

compensator utilizes the internal signals e(k)and ud(k)

to generate a compensation signal w(k).mdenotes the

relative degree of Pz−1. A nominal plant model ˆ

Pz−1

is adopted to approximate the plant dynamics. With

the plug-in compensator, the transfer function of the

overall controller from e(k)to ud(k)is

Call z−1=Cz−1+z−mˆ

P−1z−1Qz−1

1−z−mQz−1.(1)

The Q-ﬁlter here is designed such that 1−z−mQz−1

contains the internal model 1 −z−N:

1−z−mQz−1=1−z−N

1−αNz−N,(2)

where α∈[0,1) is a tuning factor that determines

the attenuation bandwidth of 1 −z−mQz−1. A wider

attenuation bandwidth can be achieved by choosing a

smaller α. However, to get better steady-state perfor-

mance, αis preferred to be close to 1.

At the harmonic frequencies (ωk=k2πf0Ts,k∈Z+,

the set of positive integers), the magnitude responses

of 1 −z−mQ(z−1) are zero because 1 −e−jωkN=1−

e−jk2πf0Ts/(f0Ts)=1−e−jk2π=0. Hence, |Call(z−1)| → ∞

and Gd→yd(z−1)=P(z−1)[1−z−mQ(z−1)]

1+P(z−1)C(z−1)=0 when z=ejωk.

At the intermediate frequencies, Q(e−jω)≈0, and |1−

z−mQ(z−1)|z=ejω≈1 when αis close to 1; thus Call(z−1)≈

C(z−1), and the original loop shape is maintained.

During implementation, a zero-phase lowpass ﬁl-

ter q0z−1is incorporated into Qz−1for robustness

against plant uncertainties at high-frequency regions:

Qz−1=1−αNz−(N−m)

1−αNz−Nq0z−1,(3)

and one such example is

q0z−1=1+z−1n0(1+z)n0

4n0,(4)

where n0∈Zis the number of the added zero pairs at

the Nyquist frequency. The cut-oﬀfrequency of Qz−1

can be additionally adjusted by adding extra zero-

phase pairs [3].

B. Cross-channel disturbances in laser-based AM

A dual-axis galvo scanner platform in Fig. 2 is a key

component in laser-based AM for laser path planning.

Typically, it is composed of two sets of mirrors, motors,

and control systems, that is, an X channel and a Y

channel. With the collaborative rotation of the two

mirrors, the input laser beam is reﬂected to generate

a predeﬁned scanning trajectory at high speed with

high precision. The rotation angles of the mirrors are

measured by encoders mounted coaxially with the

motor shaft in the scanner enclosure.

The collaborative control of the two channels intro-

duces periodic disturbances. The mechanical motion of

one rotating mirror can transmit to the other mirror. In

addition, high currents in the ground lines of the two

servo drivers can cause the channels to crosstalk [22].

When actuating one channel with a simple harmonic

signal at f0, the FFT of the non-actuated channel output

4832

Power supply

Monitor

Servo driver x

Servo driver y

ADC

ADC

DAC

DAC

Laser source

Galvo scanne r

White scr een

Server with

dSPACE and

Matlab

Fig. 2. Schematic of the hardware platform.

is observed to contain a frequency spike at f0caused

by mechanical vibrations and frequency spikes at 2n f0

(n∈Z+) due to crosstalk. Numerically, if the Y channel

is driven with a sine wave at 600 Hz and the X channel

has no input, frequency spikes at 1200Hz, 2400Hz,

3600Hz, etc arise at the FFT of the X output when the

crosstalk is unaccounted for.

III. Proposed fractional-order RC algorithms

We propose three RC algorithms to tackle a non-

integer Nin the internal model. Two indirect schemes, a

wide-band RC and a quasi RC, are ﬁrst explored. Then

we develop the analyses and applications of the main

new multirate RC.

A. Wide-band and quasi repetitive control

The wide-band and quasi RC algorithms are variants

of the plug-in compensator in (1) and Fig. 1. Both

of them are implemented at the baseline sampling

rate of Ts. In the wide-band RC, Nis rounded to

the nearest integer. To insert high control gains at the

desired harmonic frequencies, the proposed wide-band

RC increases the attenuation width of each frequency

notch in 1 −z−mQz−1, as shown in Fig. 3. With α

decreasing from 0.99 to 0.8, the magnitude response

of 1 −z−mQz−1decreases from −2.8dB to −14.2dB at

1200Hz, yielding increased control eﬀorts at the aimed

frequencies. Yet, as the attenuation width is increased

further, the ampliﬁcation of intermediate frequencies

due to the waterbed eﬀect also increases [23]. Ad-

ditional design considerations are therefore necessary

for achieving better performance, such as the gain

scheduling and the use of a damped notch ﬁlter [24].

In the proposed quasi RC, Nis designed to be

an integer by shifting the fundamental frequency to

the least common multiple (LCM) between itself and

the sampling frequency, that is, N=fs/LCMfs,f0.

The resulting new ﬁctitious fundamental frequency

f0

sis fs/N. For example, with Ts=1/16ms and f0=

1200Hz, N=16000/LCM (16000,1200)=40, and f0

s=

16000/40 =400Hz; high-gain control signals are gen-

erated at {400iHz}(i∈Z+). The desired harmonics

at {1200iHz}(i∈Z+)are thus covered in the quasi

RC with extra servo enhancement at frequencies

{400iHz}(i,k∈Z+andi,3k), as shown in the dotted

line in Fig. 4.

-20

-10

0

Magnitude (dB)

1-z-mQ(z -1)

102103

Frequency (Hz)

-80

-60

-40

-20

0

Magnitude (dB)

Q(z-1 )

=0.8

=0.95

=0.99

Fig. 3. Example magnitude responses of 1 −z−mQz−1and Qz−1

in wide-band RC.

100101102103104

Frequency (Hz)

-80

-60

-40

-20

0

Magnitude (dB)

Q(z-1 )

Multirate RC

Quasi RC

-30

-20

-10

0

Magnitude (dB)

1-z-mQ(z -1)

Fig. 4. Example magnitude responses of 1 −z−mQ(z−1) and Q(z−1)

in quasi and multirate RCs.

The stability conditions for the wide-band and quasi

RCs are the same as those for the plug-in RC design in

[24], that is, ˆ

P−1z−1,Qz−1,and Cz−1belong to the

set of stable, proper, and rational transfer functions.

The wide-band and quasi RCs are indirect solutions,

only partially attenuating the disturbances. For appli-

cations that require stronger servo enhancement, we

propose more fundamental structural changes in the

controller design, as shall be discussed next.

B. Multirate repetitive disturbance attenuation

The proposed multirate RC directly addresses the

fractional-order period by introducing a second sam-

pling frequency f0

s. Let f0

sequal the greatest common

divisor (GCD) of the sampling and the fundamental

periodic frequencies, namely, f0

s=GCDfs,f0. Then

N=f0

s/f0is an integer.

The proposed multirate RC (Fig. 5) is conﬁgured by

inserting the upsampling and downsampling blocks

into Fig. 1 before and after the all-stabilizing controller.

The solid and dashed lines represent the slow and fast

signals sampled by Tsand T0

s=1/f0

s, respectively. T0

s

and Tsare related by T0

s=Ts/F(F>1 and F∈Z+).

The transfer functions inside the Call z−1block are

implemented at T0

s.

Based on multirate signal processing (see the Ap-

pendix), the frequency response of the open-loop trans-

fer function from ydto the summing junction before

Pz−1is

4833

Cdh(z−1)↓FP(z−1)↑F

Q(z−1)

z−mˆ

P−1(z−1)z−m

d(k)

+

r(k)+e

+

udh(k)ud

+

yd

ydh(k)

−

+

w(k)

+

+

Call(z−1)

Fig. 5. Block diagram of multirate RC.

˜

Ce−jΩTs=1

F

F−1

X

k=0

Call e−jΩT0

s−2πk

F.(5)

Thus, when the reference r(k) is zero (i.e., in regula-

tion problems), block diagram manipulations in Fig. 5

give the Fourier transform of the plant output yd(k):

Yde−jΩTs=Pe−jΩTsDe−jΩTs

1+Pe−jΩTs˜

Ce−jΩTs(6)

=Pe−jΩTsDe−jΩTs

1+1

FPe−jΩTsPF−1

k=0Call e−jΩT0

s−2πk

F.

For example, let Ts=1/16ms and f0=1200 Hz. The

multirate RC gives T0

s=1/GCD(16000,1200)=1/48 ms.

The plug-in compensator is designed under the sam-

pling time of T0

ssuch that small gains of 1 −z−mQz−1

are generated at the harmonic frequencies of Ω0=2πn×

1200rad/s (n∈Z+), as shown in the solid line of Fig. 4.

For disturbances at Ω0, since the repetitive controller

gives Call e−jΩ0T0

s→ ∞, we have the summation form

of Call in (5) also goes to inﬁnity. Thus, Yde−jΩTsin

(6) converges to zero, yielding yd(kTs)=0 at Ω0.

When there is (stable and bounded) model uncer-

tainty ∆z−1such that ˆ

Pz−1=Pz−11+ ∆ z−1,

standard robust-stability analysis gives that the closed-

loop system of the multirate RC algorithm is stable if

and only if both of the following hold:

1) Nominal stability condition is satisﬁed, that is, the

closed loop is stable when ∆z−1=0.

2) Robust stability requirement is met by ap-

plying the small gain theorem [25]: for any

Ω,∆e−jΩTsTe−jΩTs<1, where Tz−1=

P(z−1)C(z−1)

1+P(z−1)C(z−1)is the complementary sensitivity

function.

The second condition translates to ∆e−jΩTs<

1/Te−jΩTs. Thus, 1/Te−jΩTsspeciﬁes the upper

bound of the plant uncertainty at all frequencies. Fig.

6 shows the magnitude responses of 1/Tz−1from

the example in Section IV. We can see that compared

with the baseline PID control, the introduction of the

multirate RC extensively preserves and increases the

robust stability bounds, especially at high frequen-

cies. With the multirate RC (solid line), the minimal

1/Te−jΩTsis −2.9 dB at 1203 Hz, which requires the

magnitude response of the uncertainty not to be greater

than 71.6% of the magnitude response of the plant at

this frequency. Without the lowpass ﬁlter q0z−1in

multirate RC (dash-dot line), the minimal 1/Te−jΩTs

decreases to −10.6 dB (29.5%) at 4801 Hz.

100101102103104

Frequency (Hz)

-10

0

10

20

30

Magnitude (dB)

Multirate RC with no lowpass filter q0 in Q

Multirate RC

Baseline control

Fig. 6. Magnitude responses of 1/Tz−1, which specify the upper

bounds of the plant uncertainties to keep robustness.

IV. experimental verification in a dual-axis galvo

scanner platform

This section provides implementation guidance and

performance comparison of the proposed algorithms

for rejecting fractional-order periodic disturbances in

the dual-axis galvo scanner platform in Fig. 2.

To attenuate the crosstalk-induced disturbances with

fractional-order periodicity in the X channel, the wide-

band, quasi, and multirate RCs in Section III are imple-

mented on top of the baseline X-channel controller.

The identiﬁed plant model with the sampling time

T0

s=1/48ms is

P0z−1=0.061z2+0.103z+0.061

z5−1.485z4+1.032z3−0.433z2−0.057z−0.061 .(7)

The plant model under the sampling time Ts=

1/16ms is:

Pz−1=0.061z4+0.737z3+0.351z2+0.034z+0.0001

z5+0.144z4−0.773z3−0.359z2−0.034z−0.0001 .(8)

Fig. 7 shows the frequency responses of the measured

and identiﬁed P0z−1of the X channel in (7).

-100

-50

0

50

Magnitude (dB)

measured system

identified system

102103104

Frequency (Hz)

-200

0

200

Phase (deg)

Fig. 7. Bode plot of P0z−1sampled at T0

s.

The plant models already contain a factory built-in

PID-type controller. The baseline feedback loop (Fig.

1 without the plug-in compensator) is thus designed

under the sampling time of Ts=1/16ms by applying

Pz−1in (8) and by letting Cz−1=1. Such a design

provides a 4400Hz bandwidth in Tz−1. Throughout

this section, n0in the low-pass ﬁlter (4) is chosen to be

3, based on the robust stability condition.

4834

The control schemes are implemented on a dSPACE

DS1104 processor board. Without loss of general-

ity, the Y channel is run with a harmonic signal

Asin2πf0t+φwith A=8 V, f0=600 Hz, and φ=0. The

periodic disturbance emerges on the X-channel output,

with frequency spikes at 1200Hz, 2400Hz, 3600 Hz, etc

(the top plot in Fig. 8). The baseline PID-type controller

is unable to reduce the crosstalk between the X and Y

channels (the dotted line in Fig. 9).

0 1000 2000 3000 4000 5000 6000 7000 8000

0

2

4

Spectrum amplitude

10-3

Baseline control 3 = 0.013159

0 1000 2000 3000 4000 5000 6000 7000 8000

0

2

4

10-3

Wide-band RC 3 = 0.0074306

0 1000 2000 3000 4000 5000 6000 7000 8000

0

2

4

Spectrum amplitude

10-3

Quasi RC 3 = 0.0070742

0 1000 2000 3000 4000 5000 6000 7000 8000

Frequency(Hz)

0

2

4

10-3

Multirate RC 3 = 0.0044765

44%

46%

66%

Fig. 8. FFT of yd(t)with the sampling of Ts.

The wide-band and quasi RC algorithms are both

implemented at the sampling time of Ts=1/16ms.

The relative degree of Pz−1in (8) equals 1 in these

conﬁgurations, that is, m=1. In the wide-band RC, N=

round1

Tsf0=round40

3=13. Under this conﬁguration,

the frequency spikes the plug-in compensator targets

are at 1

TsN=1230.77Hz and its integer multiples. A

wider attenuation width is needed in 1 −z−mQz−1to

include 1200Hz, 2400 Hz, 3600 Hz, etc. To achieve this

goal, αis set as 0.8, as shown in Fig. 3.

In the quasi RC, the ﬁctitious fundamental frequency

is conﬁgured at 400Hz such that N=16000

400 =40. Thus,

the plug-in compensator generates high-gain control

signals at 400Hz and its integer multiples, covering the

target frequencies at {1200iHz}(i∈Z+).αis designed

to be 0.999 to achieve good steady-state performance.

The time-domain result of the quasi RC is similar to

the wide-band RC result shown in Fig. 9 and is thus

omitted here. The outputs in the time domain show

that the wide-band and quasi RCs work better than

the baseline control. The frequency-domain analysis in

Fig. 8 shows the same result, and the ﬁrst two fre-

quency spikes are remarkably reduced compared with

the baseline control. Due to the decreased control eﬀort

at high frequencies, the frequency spikes at 4800Hz and

2 2.0005 2.001 2.0015 2.002 2.0025 2.003 2.0035 2.004 2.0045 2.005

-0.01

-0.005

0

0.005

0.01

yd(t)(V)

Wide-band RC Baseline control

2 2.0005 2.001 2.0015 2.002 2.0025 2.003 2.0035 2.004 2.0045 2.005

Time(sec)

-0.01

-0.005

0

0.005

0.01

yd(t)(V)

Multirate RC Baseline control

Fig. 9. yd(t)of baseline control, wide-band RC, and multirate RC.

6000Hz remain largely unchanged.

In the multirate RC, the plug-in compensator is

designed at T0

s=1/48ms with F=Ts/T0

s=3. The relative

degree of P0z−1in (7) is 3 (m=3). αis chosen to

be 0.999 to reduce the waterbed eﬀect. In this case,

N=1

T0

sf0

=48000

1200 =40. In the solid line of Fig. 4, small

gains of 1−z−mQz−1are exactly generated at 1200 Hz

and its integer multiples. The increased control eﬀorts

at high frequencies yield a further-attenuated output in

the time domain (Fig. 9) and in the frequency domain

(the bottom plot of Fig. 8). Besides the attenuated

low-frequency spikes, the frequency spikes at 3600Hz,

4800Hz, and 6000 Hz are obviously reduced.

The experimental results of the four control systems

are compared in Fig. 8. As a performance metric, for

each control system, the 3σvalue of the time-domain

result is provided, where σdenotes the standard devi-

ation. Compared with the baseline control, the appli-

cation of the multirate, wide-band, and quasi RC algo-

rithms decreases the 3σvalues, thereby achieving the

desired disturbance-attenuation eﬀect. In more details,

the wide-band and quasi RCs have similar performance

gains of 44% and 46%, respectively. The multirate RC

outperforms the other two by reaching a 66% decrease

of the 3σvalue.

V. Conclusion

In this paper, three repetitive control (RC) algorithms

are proposed to overcome the intrinsic limitation of

the internal model 1/1−z−Nwhen the sampling fre-

quency is not divisible by the fundamental disturbance

frequency. The proposed wide-band RC scheme widens

the attenuation widths to include the desired distur-

bance frequencies. The quasi RC method creates an

integer Nby selecting the least common multiple of the

sampling and fundamental disturbance frequencies as

the ﬁctitious fundamental frequency. The multirate RC

applies a second divisible sampling frequency, that is,

the greatest common divisor of the baseline sampling

frequency and the fundamental frequency. Experimen-

tal results verify the eﬀectiveness of all three algo-

4835

rithms, and the multirate RC algorithm outperforms

the wide-band and quasi ones by enhancing control

eﬀorts at high frequencies.

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Appendix

Proof of (5)

The downsampling operation in Fig. 5 gives

Ud(e−jΩTs)=1

F

F−1

X

k=0

Udh(e−j(ΩT0

s−2πk

F)).(9)

From the upsampling operation, the Fourier trans-

forms of ydand ydh are related by

Yd(e−jΩTs)=Ydh(e−jΩT0

s).(10)

Let r(k)=0. Note that

˜

C(e−jΩTs)=Ud(e−jΩTs)

Yd(e−jΩTs),Call(e−jΩT0

s)=Udh(e−jΩT0

s)

Ydh(e−jΩT0

s),(11)

and thus

F−1

X

k=0

Call(e−j(ΩT0

s−2πk

F))=

F−1

X

k=0

Udh(e−j(ΩT0

s−2πk

F))

Ydh(e−j(ΩT0

s−2πk

F))

.(12)

The denominators of the terms on the right side of

(12) satisfy

Ydh(e−j(ΩT0

s−2πk

F))=Ydh(e−j(Ω−2πk

Ts)T0

s)

=Yd(e−j(Ω−2πk

Ts)Ts)=Yd(e−jΩTs),(13)

where the second equal sign is due to the upsampling

in (10). Substituting (13) and (9) into (12) gives

F−1

X

k=0

Call(e−j(ΩT0

s−2πk

F))=PF−1

k=0Udh(e−j(ΩT0

s−2πk

F))

Yd(e−jΩTs)

=FUd(e−jΩTs)

Yd(e−jΩTs).(14)

Hence, according to (11),

F−1

X

k=0

Call(e−j(ΩT0

s−2πk

F))=F˜

C(e−jΩTs),(15)

which is equivalent to (5).

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