Performance of Single-Carrier Block Transmissions over Multipath Fading Channelswith MMSE Equalization

Shuichi Ohno and Kok Ann Donny TeoDept. of Artificial Complex Systems Engineering, Hiroshima University

Higashi-Hiroshima, 739-8527, JAPANe-mail: {ohno,donny}@hiroshima-u.ac.jp

Tel: +81-82-424-7682, Fax: +81-82-422-7195

Abstract— We study uncoded bit-error rate (BER) perfor-mance of single-carrier block transmissions, zero-padded (ZP)transmission and cyclic-prefixed (CP) transmission, when mini-mum mean-squared error (MMSE) equalizers are applied. Weshow analytically that the BER of ZP transmission with MMSEequalization degrades as the bandwidth efficiency increases, i.e.there is a trade-off between BER and bandwidth efficiency in ZPtransmission. It is also proven that ZP transmission outperformsCP transmission and uncoded orthogonal frequency-divisionmultiplexing (OFDM) on the average. Numerical examples areprovided to validate our theoretical findings and to compare theblock transmission systems.

Keywords: Block transmission, single-carrier transmission,OFDM, MMSE equalization

I. INTRODUCTION

Severe multipath channels often arise in high-rate digitaltransmissions, which necessitates sophisticated equalizationat the receiver. Maximum Likelihood (ML) equalization col-lects the available multipath diversity to improve bit-errorrate (BER) performance, but is computationally cumbersome.On the other hand, linear equalization often exhibits poorperformance due to inter symbol interference (ISI) resultingfrom the multipath. Block transmissions, including orthogonalfrequency-division multiplexing (OFDM), have been intro-duced and utilized to mitigate ISI.

In block transmissions, symbols are grouped into blocks.Adding sufficient number of redundant symbols to each blockremoves inter block interference (IBI), making efficient block-by-block processing available. In OFDM, a copy of the tail ofa block, which is called cyclic prefix (CP) is appended to thetop of the block. CP with inverse fast Fourier transform (IFFT)at the transmitter and FFT at the receiver leads to multicarriersystems including OFDM. OFDM renders a convolution chan-nel into parallel flat channels, which enables very simple one-tap frequency-domain equalization. However, OFDM suffersfrom i)fading in frequency domain ii) inter carrier interference(ICI) due to frequency offset between the transmitter and thereceiver; iii) high peak to average power ratio [1]. To avoidthese drawbacks, single-carrier CP transmission discards theIFFT at the transmitter. Although there is no IFFT operationat the transmitter, efficient frequency domain equalization isavailable at the receiver [2], [3]. Precoded OFDM has alsobeen proposed, e.g., in [4] and [5], where the transmitted block

Funded in part by Research Foundation for the Electrotechnology of Chubu.

is precoded to enhance the performance. Indeed, single-carrierCP transmission and OFDM can be considered as special casesof precoded OFDM [6].

Zero-padded (ZP) transmission is another class of single-carrier block transmissions [7], where it inserts redundantzeros instead of CP into each transmitted block. Sufficientnumber of guard zeros separate two consecutive receivedblocks and eliminate IBI. For ZP transmission with MLequalization or even with Zero-Forcing (ZF) equalization athigh SNR [8], full multipath diversity gain is enabled toenhance the system performance. However, with ML equal-ization, conventional single-carrier transmissions without zeroinsertions also exhibit maximum diversity [9]. Since the band-width efficiency of ZP is reduced by redundant zeros, ZPtransmission with ML equalization is inferior to conventionalsingle-carrier transmissions in terms of bandwidth efficiency.In this sense, the benefits of ZP transmission should comefrom linear processing.

CP transmission and ZP transmission enable relatively low-complexity block-by-block processing with linear ZF or mini-mum mean-squared error (MMSE) equalization at the receiver[3], [7]. Although their performances can be demonstratedby numerical simulations, it still remains unclear which isbetter in BER performance. As the block size gets larger, theperformances of both CP transmission and ZP transmissionconverge to conventional single-carrier transmissions. A trade-off between diversity and bandwidth efficiency at high SNRwas investigated by outage bound in [10]. However, therelation of BER performance and bandwidth efficiency atpractical SNR has not been fully addressed when a linearequalization is employed. Studying their BER performancesanalytically, we will answer these fundamental questions.

This paper deals with block transmissions with linear equal-ization. We first show that for every channel, the performanceof ZP transmission with MMSE equalization degrades as itsblock size increases. There exists a clear trade-off betweenBER and bandwidth efficiency. Then, it is proved that ZPtransmission with MMSE equalization always outperformsthe uncoded OFDM, which justifies the advantage of zeroinsertions. We also show that unlike ZP transmission, theaverage BER performance of CP transmission with MMSEequalization improves as the block size is doubled. We com-pare ZP transmission with CP transmission and establish thaton the average, CP transmission is inferior to ZP transmissionof the same block size. Numerical simulations are provided to

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validate our theoretical findings as well as to compare single-carrier ZP and single-carrier CP transmission.

�s(n)S/P �s(n) CP or

ZP�u(n)

P/S �

Fig. 1. A part of block transmission at the transmitter

II. BLOCK TRANSMISSIONS

We consider point-to-point wireless block transmissionsover time-flat but frequency-selective fading channels. Aschematic diagram of baseband equivalent block transmis-sions is depicted in Figure 1. The information-bearing se-quence {s(n)} is grouped into blocks s(n) = [s(Mn +1), . . . , s(Mn + M)]T of size M . To mitigate the effects offrequency-selective channels, we add M0 redundant symbolsto each information block to obtain transmitted blocks {u(n)}of size N := M + M0.

There are two major ways to insert redundant symbols: Oneis to append a copy of the last M0(≤ M) information symbolsto the top of the information block, which is called cyclicprefix (CP). At the receiver, the received signals correspondingto the CP are discarded. CP with IFFT at the transmitterand with FFT at the receiver results in orthogonal frequency-division multiplexing (OFDM). Since single-carrier CP trans-mission without IFFT/FFT exhibits better performance thanthe uncoded OFDM and precoded OFDM [11], [12], weconsider single-carrier CP transmission without IFFT/FFT,which we just call CP transmission in the following. Anotherinsertion of redundant symbols is to pad M0 zeros at the endof each block to form zero-padded (ZP) transmitted blocks,which is known as ZP transmission [7].

Our discrete-time baseband equivalent FIR channel {h(l)}has order L, and is considered linear time-invariant. At thereceiver, we assume perfect timing and carrier synchronization.We collect N noisy samples in an N×1 received vector. In CPtransmission, we remove the M0 samples which correspond tothe CP symbols. If the number M0 of redundant symbols isgreater than or equal to the channel order L, i.e. M0 ≥ L, IBIis completely eliminated, and we obtain [7]

x(n) = HMs(n) + v(n), (1)

where v(n) is additive white Gaussian noise (AWGN) withvariance σ2

vI . In CP transmission, x(n) has M entriesand HM is an M × M circulant matrix with first col-umn [h(0), h(1), . . . , h(L), 0, . . . , 0]T . On the other hand,in ZP transmission, x(n) has N entries and HM is atall N × M (truncated) Toeplitz matrix with first column[h(0), h(1), . . . , h(L), 0, . . . , 0]T .

We assume that M0 ≥ L so that the inter block interference(IBI) is completely removed and omit the block number n in(1) for notational simplicity. Assuming that perfect knowledge

of the channel is available at the receiver, let us first evaluatethe symbol mean-squared error (MSE) of minimum mean-squared error (MMSE) equalization for our model (1).

Let us define the SNR as

γ =σ2

s

σ2v

, (2)

where σ2s is the variance of s(n). The MMSE equalizer for

(1) is found to be [13, Chapter 12]

GM := γHHM (γHMHH

M + I)−1. (3)

The mth entry of the equalized output s = GMx can beexpressed as

sm = amsm + wm, (4)

where am is the mth diagonal entry of GMHM . From (3),we have E{ssH} = σ2

sγHHM (γHMHH

M + I)−1HM , whereE{·} stands for expectation. It follows that E{|sm|2} =σ2

sam, since am is the mth diagonal entry of GMHM =γHH

M (γHMHHM +I)−1HM . The combination of noise and

residual ISI, wm, in (4) is found to have zero mean andvariance given by

E{|wm|2} = E{|sm|2}−a2mE{|sm|2} = σ2

sam(1−am). (5)

Thus, the signal-to-interference-noise-ratio (SINR) for the mthsymbol is σ2

sa2m/[σ2

sam(1 − am)] = am/(1 − am).The (m, m)th entry of (γHH

MHM+I)−1, denoted as µ(M)m ,

is found to be related to am such that µ(M)m = 1 − am. We

can therefore reexpress the SINR for the mth symbol as

SINRm :=am

1 − am=

1

µ(M)m

− 1. (6)

We note that sm is affected by the other transmitted sym-bols and the noise wm is not strictly Gaussian. However, iftransmitted symbols are independent, then the Central LimitTheorem guarantees that as the block size gets larger, the noisebecomes Gaussian. Thus, we assume that wm is Gaussian.

Let us define a function f(·) in SINR to describe a bit-error rate (BER) or symbol-error rate (SER) performancefor a digital modulation with a finite constellation. For alldigital modulation schemes, if a symbol-by-symbol detectionis employed, f(·) is in general a monotonically decreasingfunction in SINR. Using f(·), we will investigate the BER orSER performance averaged over one block defined as

1M

M∑m=1

f(SINRm). (7)

Take, for example, the symbol-by-symbol hard detection ofQPSK signaling. The probability of the bit-error for the mthsymbol is given by f(SINRm) = Q(

√SINRm) [14], where

Q(·) is defined as

Q(x) =1√2π

∫ ∞

x

e−t2/2dt. (8)

As has been shown, SINRm is a function of µ(M)m . Thus, all

we have to do is to investigate the properties of µ(M)m .

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III. PERFORMANCES OF ZP TRANSMISSION AND CPTRANSMISSION WITH MMSE EQUALIZATION

We first consider ZP transmission. In this case, we have that

µ(M+1)m ≥ µ(M)

m , µ(M+1)m+1 ≥ µ(M)

m , (9)

for m = 1, . . . M . (See [15, Lemma 1] for a proof.) Sincethe BER (or SER) function is found to be a monotonicallyincreasing function in µ

(M)m , similar to [15, Theorem 1], we

can state the following theorem:Theorem 3.1: Suppose ZP transmission with MMSE equal-

ization. Let the number of padded zeros be M0 ≥ L, where Lis the channel order. Then, for every channel realization, theBER of ZP transmission is a non-decreasing function in theinformation block size M , that is,

BERzp,1 ≤ BERzp,2 ≤ · · · ≤ BERzp,∞, (10)

where BERzp,M is the BER of a ZP transmission of informa-tion block size M .

This theorem states a deterministic and universal char-acteristics of the BER performance of ZP transmission. Itshould be remarked that no specific modulations are imposed.Theorem 3.1 holds true for all digital modulations. For a fixedmodulation, the performance depends on the block size anddegrades as the block size of the information-bearing symbolsincreases.

The bandwidth efficiency of a block transmission can bedefined as

E :=M

M + M0. (11)

For a fixed M0, the bandwidth efficiency improves as theblock size M increases, while the performance degrades asproven in Theorem 3.1. There exists a clear trade-off betweenBER performance and bandwidth efficiency. Thus, one has tocarefully design the block size to obtain the target BER.

The lower limit of BER is BERzp,1, which is realized withM = 1. In this case, the SINR for MMSE equalization is∑L

l=0 |h(l)|2γ. For BPSK or QPSK modulation, the BER canbe expressed as Q((

∑Ll=0 |h(l)|2γ)

12 ), which is identical with

the BER of ML equalization with and without zero padding[9], [6], [16]. On the other hand, as M goes to ∞, BERzp,M

converges to Q(√

1/[γ∫ 1

0|H(ej2πf )|−2df + 1] − 1), where

H(z) is the channel transfer function H(z) :=∑L

n=0 hnz−n.They are found to be equal to the BER of conventional single-carrier transmissions with MMSE equalization, having (ideal)infinite length coefficients. Thus, we can conclude that ZPtransmission always outperforms conventional single-carriertransmissions if both employ MMSE equalization.

Since Theorem 3.1 holds true for every channel realizationwith order less than L+1 and the equalities in (9) hold only forspecial channels, averaging (10) over the channel probabilitydensity function, we establish that:

Theorem 3.2: Consider ZP transmission with MMSE equal-ization as in Theorem 3.1. Then, the BER of ZP transmissionaveraged over random channels is an increasing function inthe information block size M such that

BERzp,1 < BERzp,2 < · · · < BERzp,∞, (12)

where BERzp,M denotes the average BER of a ZP transmissionof information block size M .

Suppose, for example, i.i.d. Rayleigh channels. The averageBER of ML equalization is approximated at high SNR by(Gγ)−(L+1) with G a constant. The constant G can be viewedas a coding gain if the linear channel convolution is consideredas coding over complex field [6], [16]. The slope of theBER-SNR curve, L + 1, is the diversity order, which comesfrom i.i.d. Rayleigh channels. Although ZP transmission withlinear equalization has the full diversity gain at high SNR [8],Theorem 3.2 states that its advantage disappears as the blocksize increases.

Now, we deal with CP transmission. In CP transmission,HM in (1) is an M × M circulant matrix, which we denoteHcp,M . The circulant matrix Hcp,M can be diagonalized bythe DFT matrix F and its inverse (IDFT) [17, p. 202], suchthat

FHcp,MFH = diag[H(W 0M ),H(W 1

M ), . . . , H(WM−1M )],

(13)where WM := ej2π/M , [F ]m,n = (1/

√M)W−(m−1)(n−1)

M .From (13), one finds that all the diagonal entries of

(γHHcp,MHcp,M + I)−1 have the same value given by

µ(M) :=1M

M−1∑m=0

1γ|H(Wm

M )|2 + 1. (14)

Unlike ZP transmission, CP transmission does not exhibitan ordered BER performance for a fixed channel. To compareZP and CP transmissions, let us consider CP transmission ofblock size M and ZP transmission of block size M − L. Weprove in Appendix that

µ(M) ≥ µ(M−L)m , for all m ∈ [1,M − L]. (15)

where µ(M−L)m is the (m, m)th entry of (γHH

M−LHM−L +I)−1 of ZP transmission. In (15), the equality holds if andonly if the channel is a pure delay, that is, H(z) = cz−l forc �= 0. Since the BER function f(1/µ − 1) increases with µ,we can conclude that:

Theorem 3.3: Consider CP transmission of block size Mand ZP transmission of block size M −L. If MMSE equaliza-tion is used, the BER of CP transmission is not smaller thanthe BER of ZP transmission, i.e.

BERcp,M ≥ BERzp,M−L, (16)

where the equality holds if and only if the channel is a puredelay.

We have evaluated the performance in terms of the SNRγ. For a fixed SNR, Es/N0 = γ for ZP transmission andEs/N0 = (M + M0)γ/M for CP transmission, where Es

denotes the energy per symbol and N0 = σ2v . If we set the

same Es/N0 for ZP and CP transmission, the value of γ ofCP transmission becomes smaller than the value of γ of ZPtransmission. In other words, to keep the same value of γ,additional power in Es is required for CP transmission. Thismeans that (16) holds true not only in γ but also in Es/N0.

Since CP transmission always exhibits better performancethan the uncoded OFDM [12, Theorem 2], Theorem 3.3 alsoleads to

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Theorem 3.4: For any block size M , ZP transmission withMMSE equalization always outperforms the uncoded OFDMtransmission.

Assuming that the noise in (4) is Gaussian, the BER canbe (approximately) evaluated from a linear function of theQ functions, which is specific to the underlying modulationscheme. Suppose QPSK modulation. Then, the BER is givenby Q(

√1/µ(M) − 1). Since µ(M) lies between 0 and 1,

Q(√

1/µ(M) − 1) is a strictly convex function in µ(M) [12,Lemma 4].

The BER performance of the CP transmission of block size2M is the function of µ(2M), which can be expressed as

µ(2M) =12

(µ(M) + µ(M)

), (17)

where µ(M) = 1M

∑M−1m=0

1γ|H(W2M W m

M )|2+1 . It follows fromthe convexity of Q function that

BERcp,2M ≤ 12

[BERcp,M + Q(

√1/µ(M) − 1)

]. (18)

Let us denote the probability density function of the channelas p(h0, h1, . . . , hL). Since the equality in (18) holds only forspecial channels, averaging (18) over random channels resultsin

BERcp,2M <12

BERcp,M +

12

∫Q(

√1/µ(M) − 1)p(h0, h1, . . . , hL)dh0dh1 · · · dhL,

where BERcp,2M is the average BER of CP transmission ofblock size 2M . Changing variables hl = W−l

2Mhl for l ∈ [0, L],one finds that∫

Q(√

1/µ(M) − 1)p(h0, h1, . . . , hL)dh0dh1 · · · dhL

= BERcp,M.

This proves that:Theorem 3.5: Consider CP transmission, where the number

of CP meets M0 ≥ L. If the BER performance is expressedby (7) and f(·) is concave, then the BER of CP transmissionaveraged over random channels satisfies

BERcp,2M < BERcp,M , (19)

where BERcp,M denotes the average BER of a CP transmis-sion of information block size M .Interestingly, unlike ZP transmission, the BER performanceof CP transmission with MMSE equalization improves bydoubling the block size.

Combining Theorem 3.2, Theorem 3.3 and Theorem 3.5,we have for M ≥ L that

BERzp,M < BERzp,2M−L < BERcp,2M < BERcp,M , (20)

which leads to:Theorem 3.6: Consider ZP transmission and CP transmis-

sion with MMSE equalization. Suppose M ≥ L and M0 ≥ L.If the BER performance is expressed by (7) and f(·) isconcave, then for any block size M , the average BER of ZP

0 5 10 1510

−6

10−5

10−4

10−3

10−2

10−1

100

Eb/No

BE

R

MLZP 8ZP 16ZP 64CP 32CP 64OFDM 64

Fig. 2. BER comparison for a fixed channel of order 3

transmission is always smaller than the average BER of CPtransmission, i.e.

BERzp,M < BERcp,M , for any M . (21)This theorem gives an answer to our question: On the average,ZP transmission outperforms CP transmission. This justifiesthe advantage of zero insertions. But, as the block sizegets larger, their difference decreases and converges to zero,since BERzp,M increases (cf. Theorem 3.2), while BERcp,M

decreases at least for M = 2K with integer K (cf. Theorem3.5).

IV. NUMERICAL EXAMPLES

To validate our theoretical findings, we test ZP transmissionand CP transmission with MMSE equalization, abbreviated byMMSE-ZP and MMSE-CP, with different block sizes for afixed channel and for random channels, where the number ofredundant symbols is set to be 8. Performance of uncodedOFDM with 64 subcarriers is evaluated when hard-detectionwas used at their corresponding zero-forcing equalizer outputs.The information symbols are drawn from QPSK constellation.

The fixed channel is of order 3, whose coefficientsare [0.5957 + 0.0101i,−0.3273 − 0.3472i,−0.2910 −0.0533i, 0.1285 − 0.5599i]. Figure 2 shows the BER perfor-mance as a function of Eb/N0, where ML stands for theBER of a matched filter, which is the performance limit withtrue ML detection of any block sizes. It is clear from thefigure that the performance of MMSE-ZP degrades as itsinformation block size increases, which is analytically provenin Theorem 3.1. MMSE-CP does not have such a property.Indeed, MMSE-CP of block size 64 has better performancethan MMSE-CP of block size 32. But remember that thisdoes not always hold true for a fixed channel. Even with thesame block size, MMSE-ZP outperforms MMSE-CP, whileTheorem 3.3 assures that MMSE-ZP of block size M − Loutperforms MMSE-CP of block size M . As suggested byTheorem 3.4, MMSE-ZP exhibits better performance than theuncoded OFDM for all range of Eb/N0.

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0 5 10 15 2010

−5

10−4

10−3

10−2

10−1

100

Eb/No

BE

R

MLZP 8ZP 16ZP 64CP 32CP 64OFDM 64

Fig. 3. Average BER comparison for Rayleigh channels of order 7

To verify Theorem 3.2 and Theorem 3.5, we generated104 Rayleigh distributed channels of order L = 7, havingcomplex zero-mean Gaussian taps with exponential powerprofile: E{|h(l)|2} = exp(−l)/[

∑l exp(−l)], and averaged

the results. The BER curves are depicted in Figure 3. Wecan verify that as the block size increases, the performanceof MMSE-ZP degrades, while the performance of MMSE-CPimproves as the block size doubles, as suggested by Theorem3.2 and by Theorem 3.5, respectively. For moderate block sizeof 64, there is no significant difference between MMSE-ZPand MMSE-CP. Their performance limits, i.e. at M infinity,lie between the BER curves of MMSE-ZP with M = 64 andMMSE-CP with M = 64.

V. CONCLUSIONS

We have studied two major single-carrier block transmis-sions, CP transmission and ZP transmission, proposed tomitigate multipath fading effects. We have shown that whenMMSE equalization is adopted, the instantaneous and theaverage BER performance of ZP transmission degrade as theblock size increases, while the average BER performanceof CP transmission improves as the block size doubles. Wehave also proved that on the average, ZP transmission withMMSE equalization outperforms both CP transmission withMMSE equalization and uncoded OFDM, which highlightsthe advantage of ZP transmission with MMSE equalization.

APPENDIX

Let us define

Rcp,M = γHHcp,MHcp,M + I, (22)

Rzp,M = γHHzp,MHzp,M + I, (23)

where Hzp,M stands for the channel matrix of ZP transmissionof block size M .

It is easy to check that Rcp,M can be expressed as

Rcp,M =[

Rzp,M−L C

CH RL

], (24)

where RL is an L × L matrix and C is an (M − L) × Lmatrix. Using the block matrix inversion lemma, we find that

R−1cp,M =

[R−1

zp,M−L 00 0

]+

[−CIL

]∆−1

[−CIL

]H, (25)

where C = R−1zp,M−LC and ∆ = RL − CHR−1

zp,MC.The matrix ∆ is the Schur complement of Rzp,M−L in

Rcp,M . If Rzp,M−L is positive definite, then the number ofzero eigenvalues of ∆ equals the number of zero eigenvaluesof Rcp,M . Hence, if Rcp,M is positive definite then so is ∆.

It follows from (25) and ∆ > 0 that at least one diagonalentry of the second term of the right hand side of (25) is greaterthan or equal to zero. Noting that all the diagonal entries ofR−1

cp,M has the same value µ(M) and comparing the first M−L

diagonal entries in (25) leads to µ(M) ≥ µ(M−L)m , for all m ∈

[1,M −L]. Since Rcp,M is circulant, one finds that C = 0 ifand only if the channel is a pure delay. Thus, µ(M) = µ

(M−L)m

only when the channel is a pure delay.

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