### Frequency-Domain Interpolation-Based Channel

### Estimation

### in

### Pilot-Aided OFDM Systems

Pei-Yun

### Tsai

and Tzi-Dar Chiueh Graduate Institute of Electronics Engineeringand Department of Electrical Engineering National Taiwan University, Taipei, Taiwan, 10617.

Abstract-In this paper, we propose a novel freqneucy- domain interpolation algorithm for channel estimation in comb-type pilot-aided orthogonal frequency-division multiplex- ing (OFDM) systems. There exist two major types of pilot- aided OFDM channel estimation methods: time-domain and frequency-domain. We show that these two estimation meth-

* ods *have mathematical equivalence. The performance of these
channel estimators mainly depends on how the channel impulse
mponse

**(CIR) is**reconstructed from frequency-domain channel mponse sample at the pilot suh-carriers. By closely examining timedomain CIR characteristics, we propose a new frequency- domain interpolation-based algorithm that can overcome the limitation of conventional frequency-domain algorithms. Mean- while, this new algorithm

**has**the advantages of less latency and computation complexity when compared to time-domain approaches. Simulation

**results**show that the proposed algorithm outperforms frequency-domain interpolation-based and time- domain algorithms in most cases.

I. INTRODUCTION

OFDM has attracted considerable attention since last decade, mainly because its substantial advantages in high- rate transmission over frequency-selective fading channels. By dividing

### a

wide-band frequency-selective-fading channel into a large number of narrow-band flat-fading sub-channels, OFDM systems can easily compensate adverse channel ef- fects by a simple one-tap frequency-domain equalizer. More- over, with a cyclic prefix, inter-symbol interference can be mitigated and the mutually-overlapped spectra improve the spectrum efficiency. These features facilitate the adoption of OFDM in communications standards, such as digital au- dio broadcasting (DAB), digital video broadcasting-terresua1 (DVB-T), and IEEE 802.11dg wireless LAN.In OFDM technology, high-rate transmission is achieved by using higher-order constellations. Robust coherent detection of such OFDM signals calls for accurate channel estimation. To this end, pilot sub-caniers are often interlaced with data sub-carriers. The comb-type pilot insertion has been shown to be suitable for channel estimation in fast fading channels [I]. Various comb-type pilot-aided channel estimation schemes for OFDM systems have been proposed. Among them, there are two major types: time-domain windowing and frequency- domain interpolation.

In the time-domain windowing algorithms, a time-domain CIR is obtained by first inverse Fourier transforming the

frequency-domain channel response at the pilot sub-carriers.
In this case, the number of pilot sub-caniers *M * must be
greater than the normalized maximum excess delay, i.e.,

**A4 **

### >

*T,,,~&*[21, where

*T ~ , , ,*and

**I", **

represent the maxi-
mum excess delay and the sample time. Thereafter, different
windowing techniques are applied to the contaminated time-
domain CIR in order to reduce noise and aliasing effect. In
**I",**

* [ 3 ] , the CIR is directly cut *off below a threshold. Similarly,
Minn [4] and Fukuhara

*keep only the more significant samples in the CIR. Garacia [6l*

**[ 5 ]***CIR. Yang [2] utilizes all the CIR samples and applies*

**gathers A,f samples in the****minimum mean squared error (MMSE) weighting. **

In the frequency-domain interpolation algorithms, linear in-
terpolation of the responses at the pilot sub-caniers has been
proposed to estimate the frequency-domain channel response
for all sub-caniers **[7]. **In

**[SI, **

a second-order interpolation
technique has been shown to outperform linear interpolation.
Coleri [ 11 uses a low pass filter and spline cubic interpolation.
Usually, in frequency-domain interpolation techniques, pilot
sub-caniers over-sample the channel frequency response in
the frequency domain by at least a factor ### of

two, i.e.,*&I*

*2 *

**2rmaZ/Ts**

### VI.

We can see that these two types of channel estimation
algorithms developed along different directions. In the time-
domain windowing algorithms, researchers have tried to in-
crease estimation accuracy by weighting the time-domain CIR
samples. In the frequency-domain interpolation algorithms,
higher-order polynomials are adopted to approximate the
ideal sinc interpolation regardless of the CIR characteris-
tics. Nevertheless, these two types of algorithms have their
own limitations. In the time-domain windowing algorithms,
**although discrete Fourier ****transform **

**(DFT) **

can be imple-
mented by

**fast Fourier transform (FIT) algorithms, buffers**are needed for temporary data storage. Moreover, F l T and IFFT operations amount to overhead in latency as well as complexity, which is non-existent in the frequency-domain interpolation algorithms. On the other hand, in order to get acceptable performance, the conventional frequency-domain interpolation algorithms need more pilot sub-caniers, which reduce transmission efficiency.

In this paper, we first show correspondence between these two types of channel estimation algorithms. Then, we propose

**AwGN& **

**4 ****2" **

Fig. 1. **Baseband block d i a g m of B **typical pilot-aided OFDM system.

a novel frequency-domain interpolation-based channel esti-
mation algorithm that can efficiently extract pilot sub-carrier
information **as **the time-domain methods. Furthermore, the
proposed technique strikes a balance between noise suppres-
sion, aliasing removal, and CIR preservation by effectively
shifting the time window and using a new interpolation
function.

The paper is organized as follows. In Section **11, ** the
description of the comb-type pilot-aided OFDM system is
given. Section **111 **illustrates the correspondence between
time-domain windowing and frequency-domain interpolation.
**The **proposed frequency-domain interpolation function is
presented in Section **IV. Simulation results and comparisons **

are given in Section V. Section VI concludes this paper. 11. SYSTEM DESCRIPTION

Fig. **1 **shows a typical block diagram of an OFDM system
based on pilot-aided channel estimation. The IFFT block
transforms frequency-domain data, *X k . *on the k-th subcarrier
into time-domain samples **2, ****as **

**k = - - N I Z + l **

**(1) **

where *N * is the number of total sub-carriers. In order to
* deal with inter-symbol interference (ISI), a cyclic prefix of *
Ny samples is inserted at the beginning of every symbol.
The comb-type pilot allocation is shown in Fig.

**2. Assume**that *A4 *pilot sub-carriers **are **uniformly distributed in a total
of iV sub-carriers. The pilot sub-canier spacing is D =
*N f M , *where *D is an integer. Note that in addition to N , *

transmission sub-carriers, there are reserved sub-carriers used

**as *** guard bands on both ends of the specuum. Among the Nu *
transmission sub-carriers, there are

*11.1, *

pilot sub-carriers for
channel estimation.
At the receiver, the baseband signal is first sampled to
obtain * z,. *With the cyclic prefix removed,

*is sent to the*

**2,****0 ****0 ****0 ****Q ****0 **

**0 **

**0**

**0 **

**0**

**4 **

### +

frequency

**N **

**0 **

Guard band Pilot subcarriers **0**

*Data subcarriers Fig. 2. Pilot sub-c-ers allocation.*

**0**### FFT

block for transformation to the frequency domain:**N-1**

*z, *

= *,*

**z ,**

**e-327+h7,***=*

**k**

**- N / Z**### +

1,. ,### .

### ,

**N / 2 . ( 2 )****n=0 **

Assume that the duration of the cyclic prefix is long enough
so that there is no IS1 and that the down-conversion is accurate
enough so that there is no inter-carrier interference (ICI).
Then the frequency-domain received baseband data * z k *is
given by

**zk **

= **zk**

**X k H k**### +

*(3)*

**v k ,**where **H k is the channel response on the k-th sub-carrier and ***V, *is the noise term. The frequency-domain channel response
is given by

**Hk **

= **Hk**

*(4)*

**h , . e - j 2 n G ,**where *h,- and T~ *denote the gain and delay of the r-th path

and the CIR has the form of *h ( t ) *=

### E,

**h , ( t ) .***6 ( t *

- *T J t ) ) .*From the received frequency-domain data on the pilot sub-carriers, the channel response on those sub-carriers can

**be computed as**

*=*

**H,,,,***Z,DfX,,,D,*

*m*=

*-A4,/2*

### +

**1 , . **

### .

.### ,At,

*f2.*Henceforth, the channel response on all data sub-caniers can be estimated, and the received data are equalized by the channel estimation,

**H k**" **z k ****H k **

* X k *=

*7 ,*

**k = **

- N U / 2 f 1 , . . . , O , . . . **k =**

*,N,,/2.*

**( 5 )**111. TIME-DOMAIN WINDOWING AND FREQUENCY-DOMAIN INTERPOLATION

In time-domain channel estimation algorithms, A<-point
inverse Fourier transform is applied to the *AI *pilot sub-c+er
channel responses to reconsmct the time-domain CIR, **h,: **

Due to the sub-sampling in the frequency domain,

**Ln **

is
a folded version *samples. We can usually avoid aliasing by setting*

**of the original CIR with a period of A4***A4*

### > Ny

**since CIR energy mainly appears in the first N g taps.**Usually, a time-domain window is applied on the pen-
**odic AI-sample CIR. Let us denote this window by w **=

*[w-b w-b+1 * .., *W L - b - 1 l T , * where **L is the window width **

and *b *controls its starting position. After windowing, the esti-
mates of the frequency-domain channel response are obtained
by Fourier transforming the weighted and zero-padded CIR

**as **

**L-b-1 **

**n=-b **

**(7) **
where

### <.>

denotes modulo*M . *

The equation above can be
ipterpreted **as**interpolation in the frequency domain using

*H,D *

as base points and interpolation coefficients **1471**where

*n=-b *

Similarly, in a frequency-domain interpolation algorithm, a corresponding time-domain window can be derived. For a set of J-tap interpolation coefficients, W k , the windowing vector is given by

*J D / 2 *

* R' *-

### -

*(9)*

**I{7k,&2"nk/'Y,****k = - J D / 2 + 1 **

**" - D **

**" - D**

**The fact that these two types of operations are closely related **
offers a possibility of mapping between these two types of
**algorithms. Therefore, one can design a channel estimation **
algorithm that has advantages from both the frequency-
domain algorithms and the time-domain algorithms.

IV. **NEW FREQUENCY-DOMAIN **INTERPOLATION **FOR **

CHANNEL ESTIMATION

The inverse Fourier transform of M pilot sub-canier
**frequency-domain responses generates a periodic time- **
domain signal with a period of

**MTS, **

**MTS,**

**as**shown in Fig.

**3.**If the OFDM symbol boundary acquisition is accurate, the first pulse of the CIR will he at the origin and the remaining impulse response appears in the guard interval

*[O,N,T,].*

However, energy leakage occurs in the uniformly T,-spaced
CIR due to the non-T,-spaced channel delay * T~ *[IO]. There-

fore there exist pre-cursor as well as post-cursor in the
reconstructed CIR. **A **time-domain window **is **used to preserve
the major portion of the CIR and at the same time reject the
**aliased components. It is clear from Fig. 3 that the window **
must be shifted to the right instead of centering at the origin

**as **the conventional polynomial interpolators in * F q (9). Note *
that shifting the window in the time domain is equivalent
to rotating the phase of the interpolation coefficients in the
frequency domain.

**MT, **

**Fig. **3. **Periodic CIR in the time domain **

Usually, the magnitude of the time-domain CIR decays
gradually to the right. The time-domain CIR reconstructed
by the samples of the frequency-domain channel response **is **
corrupted by noise and aliasing effect. The weighting window
must be flat over the duration where **CIR is strong *** so *that
it is not distorted [ l l ] . On the two edges of the window,
the weighting should be smaller in order to suppress noise
and aliasing effect. Moreover, smooth weighting in the time
domain entails fast-decaying magnitude in the frequency-
domain interpolation coefficients, and thus fewer of them are
needed.

In light of the above considerations, we choose the raised-
cosine function **as the frequency-domain interpolation coeffi- **

cients and set

where

*p *

is the roll-off factor; d is the time shift and it is
decided by the worst-case channel delay spread. Note that in a
short delay-spread channel, the estimation error is insensitive
to d since the major podon of the CIR will be covered by
the window with a wide range of *d.*

**V. **SIMULATION **RESULTS AND **COMPARISONS
In order to show the effectiveness of the proposed **raised- **

cosine-based frequency-domain channel estimation algorithm, we conduct simulation

### on

some practical scenarios. In the simulation. we use the typical urban channel model given in [12], which has 20 paths and a maximum excess delay of 24.28 samples at a sampling rate of 11.52**MHz.**There are 1024 sub-carriers with 29 pilot sub-carriers evenly insetted for channel estimation. The guard interval

**has**

### a

length of

**26**samples.

Fig. 4 demonstrates the improvement in channel esti- mation accuracy if we consider the time shift effect by incorporating a phase rotation term in several frequency- domain interpolation-based algorithms. Since the time- domain weighting window derived from the polynomial interpolators' coefficients has wide and non-flat mainlobe

**[13], ** aliasing **is **unavoidable. So the CIR will be distorted,
yielding larger estimation error. We can also see the proposed
raised-cosine interpolator is by far the best frequency-domain
interpolation-based channel estimator in terms of estimation
accuracy. In addition, note that for each algorithm, the ermr

**Long Delay Spread (3068) **
lo

## r-

### I

_{: I : : : : }

_{... }

### ---

'*-*

*7*

~ . . . : . . . **. . j**. . .

### ]

. . . . ~ . . .**b**. . .

**4 **

## 1

...### ;.-?@-q

. . . .-**E**.i . . . -&- Cubic ~

**. Raised-cosine.**:

**0.1**

**0.2****0.3**

**0.4**

*diM*

**0.5****Long Delay Spread **

**too **
**10 **
**a **
Y m
**to **
**SNR **
h g . **4. **

in **frequency-domain interpolation-based channel estimation. **

**Improvement in estimtioh accumcy **if **the time shift is considered ** **Fig. ****6. **

**channel with long delay spread. **

**Bit **error **rate performance using different channel estimators in ****B **

**Long Delay Spread **

100
Lu to-'
**0) ****I **
**5 ****E **
**c **
...
.-
**I **
i o - 2
**5 ** t o 15 **20 ****25 ** **30 **
**SNR **
**Fig. ***5. *

long **delay spread. **

**Estimation ermr of different channel estimators in a channel with **

decreases as the time shift increases, and after reaching a minimum, the error starts to rise again.

Fig. 5 illustrates the performance of several channel estima- tors under different levels of SNR. RC-6 and RC-8 denote the proposed frequency-domain interpolation channel estimation algorithm using 6-tap and 8-tap raised-cosine coefficients, respectively. We then apply such estimated channel response to a frequency-domain equalizer in a 16-QAM sub-canier

**data **receiver. The resulting hit error rates versus * S N R *are
depicted in Fig. 6. From these two figures, we see that in
the case of channels with long delay spread, the proposed
algorithm is significantly better than all other frequency-
domain algorithms and most of the time-domain algorithms.
Computational complexity and memory requirement are
also important issues in the choice of channel estimators .

The time-domain windowing estimators require A$ **log, AT **

### +

IV log, N

**complex multiplications and Af**### +

N - 2 complex buffers for radix-4 FFTnFFT and 2A2 real multiplications for windowing by**w**in the time domain. The entries in

**w **are pre-computed and stored thus occupy

*M *

buffers.
Due to the latency of *- 1 samples caused by IFFT*

**2N**and

### FFT,

*21%'" complex data must be also buffered. For*

the proposed frequency-domain J-tap interpolation channel
estimator (shown in Fig. *7). *we need *ZNu J *

### +

4fi-u real*multiplications and 25 real data buffers and*(D - l)J real buffers for the real coefficients. Since the latency of the frequency-domain interpolation is much reduced to JD/Z-1, only J ( D

### -

1)/2 complex data buffers are necessary.As the FFI size increases, the overhead in the buffers and multiplications increases rapidly if a time-domain windowing channel estimator is implemented. The numbers of multipli- cations and of buffers needed by the time-domain channel estimators and the frequency-domain estimators for different

### FFT

size are plotted in Fig. 8. Note that a 6-tap frequency- domain interpolation estimator is used for comparison. For the proposed estimator, the reduction in computational com- plexity is evident, especially when the number of sub-caniers is large.VI. CONCLUSION

In this paper, we examined the time-domain and frequency-
domain chaMel estimation algorithms in the comb-type pilot-
aided OFDM systems. We showed that there exists correspon-
dence between these two types of channel estimators. We then
proposed to include a phase rotation term in the frequency-
domain interpolation to account for the equivalent time shift
for better CIR window location. In addition, the proposed
raised-cosine interpolation coefficients not only have good
attenuation on noise and aliasing but **also **preserve most CIR
information and provide very good estimation accuracy. The

Fig. 7.

**channel **estimator.

Block diagram of the proposed frequency-domain interpolation

Complexity **comparison **
**106 **
**1 os **
.-

**E **

**E**

**2**

**10‘***a *

**1 o3**

**1 O2**

**1 o2**

**1**

*0’*10‘

**No. of subcamen**Fig. 8.

estimton: F.D.: frequency-domain estimators.

Channel estimtor complexity versus FFI size. T.D.: time-donuin

complexity of the proposed frequency-domain interpolation

### is

shown to be less than all time-domain channel estimators in almost all FFT sizes. With its low hardware complexity and accurate estimation performance, the raised-cosine frequency- domain interpolation channel estimator will find many appli- cations in**OFDM **

communication systems.
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