The Vault

Comparisons of Filter Bank Multicarrier Systems
Research Paper / Feb 2014

 

978-1-4577-1343-9/12/$26.00 ©2013 IEEE 

 

Comparisons of Filter Bank Multicarrier Systems 

 

 

Juan Fang1, Zihao You2, I-Tai Lu5 

ECE Department 

 

Polytechnic Institute of NYU 

Brooklyn, NY, USA 

 

jfang1985@gmail.com1, zyou01@students.poly.edu2, 

itailu@poly.edu5 

 

Jialing Li3, Rui Yang4 

InterDigital Communications, Inc. 

 

Melville, NY, USA 

Jialing.Li@interdigital.com3, Rui.Yang@interdigital.com4

 

  

Abstract—In this paper, we compare three filter bank 

multicarrier (FBMC) techniques, i.e., Orthogonal Frequency 

Division Multiplexing (OFDM), OFDM-Offset Quadrature 

Amplitude Modulation (OFDM-OQAM) and wavelet packet 

modulation (WPM), based on multiple criteria, including 

bandwidth efficiency, computational complexity, latency, out-of-

band emission, peak-to-average power ratio, and sensitivity to 

timing and frequency offsets. An efficient implementation of 

OFDM-OQAM is also proposed for a special case. 

 

Keywords-Multicarrier; OFDM; OFDM-OQAM; Wavelet 

Packet Modulation  

 

I. INTRODUCTION 

Multicarrier modulation (MCM) techniques enable 

 

transmission of a set of data over multiple narrow band 

subcarriers simultaneously. With an advanced wideband 

modulation and coding scheme (MCS), a system with MCM 

can achieve much higher spectral efficiency in frequency 

selective channels compared to those using single carrier 

modulation techniques. Filter bank multicarrier (FBMC) 

modulation is a family of MCM techniques in which a 

prototype filter is designed to achieve a certain goal, such as 

minimizing inter-symbol interference (ISI), inter-carrier 

interference (ICI) and/or stop band energy. The well-known 

Orthogonal Frequency Division Multiplexing (OFDM) can be 

considered a type of FBMC whose time domain prototype 

filter is a simple rectangular pulse. From the complexity 

perspective, the main advantage of OFDM over other FBMC 

techniques is that it is very easy to be implemented. However, 

large sidelobes of the rectangular pulse in OFDM create 

challenging issues in practical systems. For example, the 

performance of the systems at the physical (PHY) layer is very 

sensitive to frequency offset. In addition, in systems such as 

those in TV white space and heterogeneous networks with 

small cells, multiple radio links coexist in congested spectral 

bands, but are loosely controlled or coordinated in resource 

usage (frequency, timing and power). In such a network, 

strong adjacent channel interference may be generated from 

large out-of-band emissions partially contributed from large 

sidelobes at the baseband. Another drawback of OFDM is that 

the modulated signal exhibits large peak-to-average power 

ratio (PAPR) which usually leads to low efficiency power 

amplifiers (PAs). To overcome the aforementioned drawbacks, 

researchers have been looking into different FBMC techniques, 

including OQAM-OFDM [1]-[4] and Wavelet Packet 

Transform (WPT) [5]. In OQAM-OFDM, subcarriers of the 

signal overlap each other to achieve a high spectral efficiency. 

 

Different from OFDM, the real and imaginary parts of the 

QAM symbols are processed separately with 2×symbol rate. A 

prototype filter needs to be carefully designed to minimize or 

zero out ISI and ICI while keeping the sidelobes small. On the 

other hand, wavelet packet modulation (WPM), which uses the 

wavelet packet transform (WPT) instead of the discrete 

Fourier transform (DFT), is receiving growing attention 

because of some of its interesting features in terms of 

spectrum usage and signal construction flexibilities [5]. Some 

studies have also shown that the WPM signal exhibits smaller 

PAPR than OFDM [6]. Little about the out-of-band leakage 

from WPM waveforms is found in the literature.  

 

In this paper, we compare OFDM with other FBMC 

techniques including OFDM-OQAM and WPM in terms of 

several metrics that are commonly considered when designing 

a communication system. The metrics include bandwidth 

efficiency, computational complexity, latency, out-of-band 

emission, PAPR, and bit error rate (BER). Sensitivity to 

frequency and timing offsets is also measured and compared. 

An efficient implementation of the OFDM-OQAM synthesis 

and analysis filter banks when the prototype filter is of odd 

length is also proposed. 

 

II. FILTER BANK MULTICARRIER SYSTEMS 

Fig.1 shows a block diagram for a general FBMC 

 

system.The transmitted signal, x[n], can be expressed as 

 

ݔሾ݊ሿ ൌ ෍ ෍ ݏ௞,௟݃௞ሾ݊ െ ݈ܮ௦ሿ

ெିଵ

 

௞ୀ଴௟

 (1) 

 

where M is the number of subcarriers, s௞,௟ is the lth symbol in 

the kth subcarrier, Ls is the number of samples per transmit 

symbol spacing, and ݃௞ሾ݊ሿ  is the synthesis filter for the kth 

subcarrier. At the receiver, the estimated lth symbol sො௞,௟ in the 

kth subcarrier is 

 

̂ݏ௞,௟ ൌ ሺݕሾ݊ሿ כ ௞݂ሾ݊ሿሻ௡ୀ௟௅ೞ (2) 

where y[n] is the received signal and ௞݂ሾ݊ሿ  is the impulse 

response of the analysis filter for the kth subcarrier. Define the 

number of samples per symbol duration as L, where ܮ ൒ ܯ. 

Note that Ls and L are not necessarily the same. For an ideal 

channel where ݕሾ݊ሿ ൌ ݔሾ݊ሿ, a QAM symbol sො௞,௟ in OFDM and 

WPM will be the same as the input symbol s௞,௟  if the filter 

satisfies the orthogonal condition: 

 

ۃ݃௞ሾ݊ െ ݉ܮ௦ሿ, ௜݂ሾ݊ െ ݈ܮ௦ሿۄ ൌ ߜ௞௜ߜ௠௟ (3)

where ߜ௔௕ is the Kronecker delta. Define the lth input symbol 

vector along all subcarriers as  

 

ܛ௟ ൌ ሾݏ଴,௟ ݏଵ,௟ ڮ ݏெିଵ,௟ሿ், (4)

 

 

 

978-1-4577-1343-9/12/$26.00 ©2013 IEEE 

 

the corresponding output signal vector from the transmitter as 

ܠ௟ൌ ሾݔሾሺ݈ െ 1ሻܮሿ ݔሾሺ݈ െ 1ሻܮ ൅ 1ሿ ڮ ݔሾ݈ܮ െ 1ሿሿ், (5)  

 

the corresponding received signal vector at the receiver as 

ܡ௟ൌ ሾݕሾሺ݈ െ 1ሻܮሿ ݕሾሺ݈ െ 1ሻܮ ൅ 1ሿ ڮ ݕሾ݈ܮ െ 1ሿሿ், (6)  

 

and the estimated lth symbol vector as  

ܛො௟ ൌ ሾ̂ݏ଴,௟ ̂ݏଵ,௟ ڮ ̂ݏெିଵ,௟ሿ்.  (7)

 

In the following subsections, we give the synthesis and analysis 

filters for three different MCM schemes in details. 

 

[ ]kg n

 

[ ]0g n

 

[ ]1Mg n−

 

0,s l

 

1,sM l−

 

,sk l

 

0,ˆ ls

 

1,sˆM l−

 

,sˆk l[ ]kf n

 

[ ]0f n

 

[ ]1Mf n−

 

[ ]x n [ ]y n

 

Figure 1. General structure for an FBMC system. 

 

A. OFDM 

In OFDM, the input symbols ݏ௞,௟  are QAM symbols and 

 

ܮ௦ ൌ ܮ. The synthesis and analysis filters are defined as 

݃௞ሾ݊ሿ ൌ ௞݂ሾ݊ሿ ൌ ݌ሾ݊ሿ ௅ܹି ௞௡  (8) 

 

where ௅ܹ ൌ ݁ି௝ଶగ/௅ . The prototype filter ݌ሾ݊ሿ ൌ 1/√ܮ  for ݊ ൌ 0,1, … , ܮ െ 1 and 0 elsewhere. Substituting (8) into (1), 

after some math manipulations, the transmitted signal for the 

lth symbol is obtained as 

 

ܠ௟ ൌ

1

 

√ܮ ۴௅

ுሾ۷ெ ૙ெൈሺ௅ିெሻሿ்ܛ௟ (9)  

 

where the (i,j)th element of the ܮ ൈ ܮ DFT matrix ۴௅ is 

ሺ۴௅ሻ௜௝ ൌ ௅ܹሺ௜ିଵሻሺ௝ିଵሻ, ݅, ݆ ൌ 1, … , ܮ (10) 

 

and the inverse DFT (IDFT) matrix is given by ۴௅ு/ܮ . ۷ெ 

stands for the ܯ ൈ ܯ identity matrix. Similarly, substituting (8) 

into (2), after some math manipulations, the estimated lth 

symbol is obtained as 

 

ܛො௟ ൌ

1

 

√ܮ ሾ

۷ெ ૙ெൈሺ௅ିெሻሿ۴௅ܡ௟  (11) 

 

The OFDM transmitter can be implemented by using zero-

padding, inverse fast Fourier transform (IFFT) and parallel-to-

serial (P/S) conversion; the OFDM receiver can be 

implemented by using serial-to-parallel conversion (S/P) 

conversion and fast Fourier transform (FFT). For multipath 

channels, cyclic prefix (CP) is usually used in OFDM. 

 

B. OFDM-OQAM 

In OFDM-OQAM, the input symbols ݏ௞,௟  are OQAM 

 

symbols and ܮ௦ ൌ ܮ/2. The OQAM symbols are defined as 

ݏ௞,௟ ൌ ݆௞ା௟ݏ௞,௟ோ  (12)

 

where ݏ௞,௟ோ  is the real input sequence: 

ݏ௞,௟ோ ൌ ൝

 

ݏҧ௞,௟′ூ , ݈ ൌ 2݈′

ݏҧ௞,௟′ொ , ݈ ൌ 2݈′ ൅ 1

 

 (13) 

 

where ݏҧ௞,௟′ூ  and ݏҧ௞,௟′ொ  are the real and imaginary parts of the ݈′th 

input QAM symbol ݏҧ௞,௟′ , respectively. In (12), the ݆௞ା௟ factor 

 

introduces the π/2 phase shift between any pair of adjacent 

OQAM symbols (in both the frequency and time) to ensure 

orthogonality. In OFDM-OQAM, (3) is not satisfied. 

According to (12), the estimated real sequence is 

 

̂ݏ௞,௟ோ ൌ ܴ݁ൣ݆ିሺ௞ା௟ሻ̂ݏ௞,௟൧ (14)

And, according to (12), the estimated ݈′th QAM symbol is  

 

ݏҧመ௞,௟′ ൌ ̂ݏ௞,ଶ௟′ோ ൅ ݆̂ݏ௞,ଶ௟′ାଵோ  (15)

As in [3], the synthesis and analysis filters are 

 

݃௞ሾ݊ሿ ൌ ௞݂ሾ݊ሿ ൌ ݌ሾ݊ሿ ௅ܹ௞௕ ௅ܹି ௞௡ (16) 

where ܾ ൌ ൫ܮ௣ െ 1൯/2, and the length-ܮ௣ prototype filter ݌ሾ݊ሿ 

is a square-root Nyquist filter with roll-off factor not greater 

than 1. The prototype filter should be designed to strike a 

balance between minimizing structure based interference, and 

channel/radio impairment mitigation. The prototype filter 

design is out of the scope of this paper and we will report it in 

the future. In this paper we use the frequency sampling based 

prototype filters [3] in our simulation. We further assume ܮ௣ ൌ

ܭܮ ൅ ݀ , where the integer K is the overlapping factor (the 

number of symbol durations in the prototype filter length), and 

݀  is an odd integer. So, ܾ  is an integer in this case. The 

synthesis and analysis filters can be implemented in different 

ways. An efficient implementation of the OFDM-OQAM will 

be presented in Sec. III. 

 

C. WPT 

WPM is a multicarrier modulation based on WPT. In WPM, 

 

the input symbols ݏ௞,௟  are QAM symbols and ܮ௦ ൌ ܮ . The 

synthesis filter ݃௞ሾ݊ሿ in (1) is derived by having the input data 

symbols go through several sub-stages where the input of each 

sub-stage first gets upsampled by 2, and then goes through a 

pair of filters ݑሾ݊ሿ  and ݒሾ݊ሿ . A 2 sub-stage (4 subcarriers) 

example is shown in Fig.2. Let ܩ௞ሺݖሻ, ܷሺݖሻ and ܸሺݖሻdenote 

the Z-transform of ݃௞ሾ݊ሿ, ݑሾ݊ሿ and ݒሾ݊ሿ, respectively. Then,  

 

ܩ଴ሺݖሻ ൌ ෑ ܷ൫ݖଶ೘షభ൯

୪୭୥మ ெ

 

௠ୀଵ

 

ܩଵሺݖሻ ൌ ܸ ቀݖଶౢ౥ౝమ ಾషభቁ ෑ ܷ൫ݖଶ೘షభ൯

୪୭୥మ ெିଵ

 

௠ୀଵ

, …, 

 

ܩெିଵሺݖሻ ൌ ෑ ܸ൫ݖଶ೘షభ൯

୪୭୥మ ெ

 

௠ୀଵ

 

 

 

 

 

(17)  

 

 

Figure 2. Block diagram of a WPM transmitter example with M = 4. 

 

If an additive white Gaussian noise (AWGN) channel is 

assumed, the structure of the receiver is simply the reverse of 

the transmitter, where the analysis filter ௞݂ሾ݊ሿ  in (2) is 

determined by another pair of half-band low-pass filter ݑොሾ݊ሿ 

and half-band high-pass filter ݒොሾ݊ሿ, respectively, as shown in 

Fig.3 (4 sub-carriers, i.e., 2 sub-stages, are drawn in this figure). 

These two pairs of filters are jointly designed so that (3) is 

 

ݑሾ݊ሿ↑2

 

ݒሾ݊ሿ↑2 

 

ݑሾ݊ሿ↑2 

 

ݒሾ݊ሿ↑2

 

↑2 

 

↑2 

 

ݑሾ݊ሿ

 

ݒሾ݊ሿ

 

ݔሾ݊ሿ

 

ݏ଴,௟

 

ݏଵ,௟

 

ݏଶ,௟

 

ݏଷ,௟

 

 

 

978-1-4577-1343-9/12/$26.00 ©2013 IEEE 

 

satisfied. In this paper, we use Daubechies (Db) wavelets [7] of 

different lengths as ݑሾ݊ሿ, ݒሾ݊ሿ, ݑොሾ݊ሿ and ݒොሾ݊ሿ. 

 

 

Figure 3. Block diagram of a WPM receiver example with M = 4 

 

III. EFFICIENT IMPLEMENTATION OF OFDM-OQAM 

Efficient polyphase implementations of OFDM-OQAM 

 

synthesis filter bank (SFB) and analysis filter bank (AFB) have 

been investigated in [1]-[4] (IFFT-based SFB and AFB for 

ܮ௣ ൌ ܭܮ ൅ 1 [1] and for general ܮ௣ [4], IFFT-based SFB and 

FFT-based AFB [2]-[3]). To make a system operable using 

with either OFDM or OFDM-OQAM, the IFFT-based SFB and 

FFT-based AFB should be chosen. It has been shown in [2]-[3] 

that the complexity of such implementation for the special case 

of ܮ௣ ൌ ܭܮ െ 1  has lowest complexity. In this section, we 

extend it to a novel efficient implementation for the case when 

ܮ௣ is an arbitrary odd integer.  

A. Synthesis Filter Bank (SFB) 

 

The Z-transform’s of the synthesis filters are 

ܩ௞ሺݖሻ ൌ ௅ܹ௞௕ൣ1 ௅ܹି ௞ ڮ ௅ܹି௞ሺ௅ିଵሻ൧ۯሺݖ௅ሻ܋ሺݖሻ (18) 

 

where  

ۯሺݖ௅ሻ ൌ ݀݅ܽ݃ሼܣ଴ሺݖ௅ሻ ܣଵሺݖ௅ሻ ڮ ܣ௅ିଵሺݖ௅ሻሽ (19) ܋ሺݖሻ ൌ ሾ1 ݖିଵ ڮ ݖିሺ௅ିଵሻሿ். (20)

 

The polyphase filters in (19) are defined as 

ܣ௤ሺݖሻ ൌ ෍ ݌ሾݍ ൅ ݉ܮሿݖି௠௠ , ݍ ൌ 0,1, ڮ , ܮ െ 1. (21) 

 

Define the Z-transform of the input symbols ܛ௟  (the Z-

transform is performed along the symbol index l) as ܁ሺݖሻ . 

Similar to [3], the Z-transform of ݔሾ݊ሿ can be written as 

 

ܺሺݖሻ ൌ ܋்ሺݖሻۯሺݖ௅ሻ۴௅ு઩గ௕ሾ۷ெ ૙ெൈሺ௅ିெሻሿ்܁ ൬ݖ

ଶ൰. (22) 

 

where 

઩గ௕ ൌ ݀݅ܽ݃൛1 ௅ܹ௕ ڮ ௅ܹሺ௅ିଵሻ௕ൟ (23) 

 

Then, ܺሺݖሻ can be further rewritten as 

ܺሺݖሻ ൌ ܋்ሺݖሻሼۯଶ۾గ௕۴௅ுሾ۷ெ ૙ெൈሺ௅ିெሻሿ்܁ሽ ൬ݖ

 

ଶ൰ (24) 

 

where 

ۯଶሺݖሻ ൌ ۯሺݖଶሻ. (25) 

 

and the permutation matrix 

 

۾గ௕ ൌ ൤

૙௕ൈሺ௅ି௕ሻ ۷௕

 

۷௅ି௕ ૙ሺ௅ି௕ሻൈ௕൨ ൌ

1

ܮ ۴௅

 

ு઩గ௕۴௅ (26)

 

From (24), the OFDM-OQAM SFB can be implemented by 

using zero-padding, IFFT, permutation ۾గ௕ , polyphase filters ܣ௤ሺݖଶሻ, and P/S conversion (via upsampling by ܮ/2 and delay 

chain). Its polyphase structure is given in Fig.4. 

 

B. Analysis Filter Bank (AFB) 

Define the Z-transform of the received signal ݕሾ݊ሿ as ܻሺݖሻ 

 

and that of the estimated symbols ܛො௟  (the Z-transform is 

 

performed along the symbol index l) as ܁෠ሺݖሻ . Since the 

analysis filters are the same as the synthesis filters, in an 

AWGN channel, the IFFT-based AFB could be written as 

 

܁෠ ൬ݖ௅ଶ൰ ൌ ሼሾ۷ெ ૙ெൈሺ௅ିெሻሿ۴௅ு۾గ௕்ۯଶሽ ൬ݖ

ଶ൰ ܋ሺݖሻܻሺݖሻ. (27) 

 

Note that the DFT and IDFT matrix could be related through 

۴௅ு ൌ ۴௅۾గ௔ (28) 

 

where ۾గ௔ is the permutation matrix  

 

۾గ௔ ൌ

ۏێ

ێێ

ۍ1 00 ڮ

 

ڮ ڮ 0

ڮ 0 1

 

0 ڮ

ڭ گ

0 1

 

0 1 0

گ گ ڭ

0 ڮ 0 ے

 

ۑۑ

ۑ

ې

. (29) 

 

Substituting (28) into (27), we obtain 

 

܁෠ ൬ݖ௅ଶ൰ ൌ ሼሾ۷ெ ૙ெൈሺ௅ିெሻሿ۴௅۾గ௖ۯଶሽ ൬ݖ

ଶ൰ ܋ሺݖሻܻሺݖሻ, (30) 

 

where the permutation matrix 

۾గ௖ ൌ ۾గ௔۾గ௕். (31) 

 

Based on (30), the FFT-based OFDM-OQAM AFB can be 

implemented by using S/P conversion (via delay chain and 

downsampling by L/2), polyphase filters ܣ௤ሺݖଶሻ, permutation 

۾గ௖ , FFT, and discarding irrelevant outputs. Its polyphase 

structure is given in Fig.5.  

 

L-point 

IFFT

 

A0(z2)

 

A1(z2)

 

A2(z2)

 

AL-1(z2)

 

↑L/2

 

↑L/2

 

↑L/2

 

↑L/2

 

Z-1

 

Z-1

 

Z-1

 

+

 

+

 

+ x[n]s0,l

 

s1,l

 

s2,l

 

sM-1,l

0

 

0

 

2

 

L-1

Polyphase 

filtering

 

P/S conversion

 

Pπb

 

b

 

b+1

 

L-1

 

0

 

1

 

b-1

 

0

 

1

 

 

Figure 4. Polyphase implementation of the OFDM-OQAM SFB 

 

 

Figure 5. Polyphase implementation of the OFDM-OQAM AFB 

 

Note that the inputs to the SFB (resp. the outputs from the 

AFB) are purely real or imaginary, and so, a L/2-point cosine 

modulated filter bank and a L/2-point sine modulated filter 

bank could be used to replace the IFFT at the SFB (resp. the 

FFT at the AFB), as in the critically sampled exponentially 

modulated filter bank [8]. As such, the AFB’s complexity 

could be reduced almost by half. However, if the system 

design strategy is to make it operable using either OFDM or 

 

ݑොሾ݊ሿ ↓2

 

ݒොሾ݊ሿ ↓2 

 

ݑොሾ݊ሿ ↓2 

 

ݒොሾ݊ሿ ↓2 

 

ݕሾ݊ሿ 

 

̂ݏ଴,௟

 

̂ݏଵ,௟

 

̂ݏଷ,௟

 

̂ݏଶ,௟

 

↓2 

 

↓2 

 

ݑොሾ݊ሿ 

 

ݒොሾ݊ሿ 

 

 

 

978-1-4577-1343-9/12/$26.00 ©2013 IEEE 

 

OFDM-OQAM, the IFFT-based SFB and FFT-based AFB 

structure should be chosen. 

 

IV. SYMBOL DENSITY, COMPLEXITY AND LATENCY 

 

A. QAM Symbol Density 

Define the time and frequency spacings for transmitting one 

 

QAM symbol (equivalently, two PAM symbols) as TQAM and 

FQAM, respectively. The bandwidth efficiency [9] in terms of 

QAM symbol/sec/Hz, is defined as  

 

ߛ ൌ 1 ൫ ொܶ஺ெܨொ஺ெ൯⁄  (32)

Denote one symbol duration as ܶ  (in seconds). In OFDM, 

assuming using CP, ܨொ஺ெ ൌ 1/ܶ, and ொܶ஺ெ ൌ ܶ ൅ ஼ܶ௉  where 

 

஼ܶ௉  is the CP duration. In OFDM-OQAM, ܨொ஺ெ ൌ 1/ܶ, and 

ொܶ஺ெ ൌ ܶ. In WPM [10], for each sub-stage of the transmitter, 

 

the time resolution is doubled and the frequency resolution is 

halved. Therefore the product of these two values remains 1. 

The QAM symbol densities of the three FBMC systems are 

summarized in Table 1. It is clear that, with CP, OFDM is less 

bandwidth efficient than OFDM-OQAM and WPM. 

 

B. Computational Complexity 

Based on the FBMCs’ efficient implementations described 

 

in the previous sections, we discuss their computational 

complexities in the critically sampled scenario, i.e., ܮ ൌ ܯ . 

We evaluate the complexity in terms of the number of real 

multiplications (but multiplications with േ1  and േ݆  are not 

included since they are flip of sign and/or flip of real and 

imaginary parts) per M QAM input symbols. 

 

For OFDM, note that the number of real multiplications of 

an ܮ -point FFT/IFFT (via Split Radix FFT [11]) with L 

complex inputs is ߤிி்஼ ൌ ܮ logଶ ܮ െ 3ܮ ൅ 4 . Therefore, the 

total number of real multiplications of OFDM per M QAM 

symbols is ߤைி஽ெ ൌ 2ߤிி்஼ ൌ 2ܮ logଶ ܮ െ 6ܮ ൅ 8. 

 

For OFDM-OQAM, note that, via Split Radix FFT [11], the 

number of real multiplications of an ܮ -point IFFT with L 

purely real or imaginary inputs is ߤிி்ோ ൌ ௅ଶ logଶ

 

ଶ െ 3

 

ଶ ൅ 4. 

 

Therefore, the total number of real multiplications of OFDM-

OQAM per M QAM symbols (equivalently, 2M PAM 

symbols) is ߤைி஽ெିைொ஺ெ ൌ 2൫ߤிி்ோ ൅ 2ܮ௣൯ ൅ 2൫ߤிி்஼ ൅

2ܮ௣൯ ൌ 3ܮ logଶ ܮ ൅ ሺ8ܭ െ 10ሻܮ ൅ 24. 

 

According to [5], the total number of real multiplications of 

WPM per M QAM symbols is ߤௐ௉ெ ൌ 4ሺܮ െ 1ሻܮ௤ , where ܮ௤ 

denotes the length of filters ݑሾ݊ሿ and ݒሾ݊ሿ. For example, the 

Dbx wavelet is of length 2x, where x is a positive integer. 

 

The complexities in terms of numbers of real 

multiplications in OFDM, OFDM-OQAM with typical K 

values, and WPM with various filters (Db1 and Db7 [7]) are 

shown in Table 2. We see that the computational complexities 

of the OFDM-OQAM and WPM could be up to 6 and 7 times 

higher than the OFDM, respectively. This more or less reflects 

the difference in implementation cost (e.g., gate and memory 

counts if implemented in ASIC) although they don’t map to 

the cost directly. However, due to the decoder complexity, the 

contribution of the MCM processor to the overall receiver 

complexity is not as significant. Therefore the increased 

complexity of the OFDM-OQAM or WPM processors relative 

 

to the OFDM processor may not be of concern when 

considered in the context of the entire baseband processor. 

 

TABLE 1 QAM SYMBOL DENSITY (SYMBOL/SEC/HZ) 

OFDM OFDM-OQAM WPM 

 

ܶ/ሺܶ ൅ ஼ܶ௉ሻ 1 1 

TABLE 2 COMPUTATIONAL COMPLEXITY 

 

ܮ ߤைி஽ெ ߤைி஽ெିைொ஺ெ (ܮ௣ ൌ ܭܮ ൅ 1) ߤௐ௉ெ 

K = 3 K = 4 K = 5 Db1 Db7 

 

128 1032 4504 5528 6552 1016 7112 

512 6152 21016 25112 29208 4088 28616 

2048 32776 96280 112664 129048 16376 114632 

 

TABLE 3 OFDM LATENCIES 

TCP 0 5.2μs 4.7μs 16.7μs 

 

߬ைி஽ெ 0.0667ms 0.0719ms 0.0714ms 0.0834ms 

TABLE 4 OFDM-OQAM LATENCIES 

 

K 3 4 5 

߬ைி஽ெିைொ஺ெ 0.3000 ms 0.3667 ms 0.4334 ms 

 

TABLE 5 WPM LATENCIES (ASSUMING ܮ ൌ ∞ሻ 

Dbx Db1 Db4 Db7 

 

߬ௐ௉ெ 0.1333 ms 0.5333 ms 0.9333 ms 

C. Latency 

 

The latency discussed in this sub-section is the inherent 

latency introduced in FBMC’s structure from the input of the 

SFB to the output of the AFB. The latency due to the 

arithmetic operations (e.g., multiplication) is ignored since 

they depend on hardware or DSP implementation. Let the 

sample duration be ௦ܶ ൌ ܶ/ܮ. In OFDM, latency comes from 

the P/S and S/P conversion pair and CP (of duration ஼ܶ௉). The 

latency due to an L:1 P/S and 1:L S/P conversion pair is 

ܮ ௦ܶ ൌ ܶ. The total latency is ߬ைி஽ெ ൌ ܶ ൅ ஼ܶ௉. 

 

In OFDM-OQAM, the prototype filter support is ൫ܮ௣ െ

1ሻ ௦ܶ ൌ ܭܶ . Latency comes from the OQAM modulation 

(which is T/2), P/S and S/P conversion pair, and filtering. The 

total latency is ߬ைி஽ெିைொ஺ெ ൌ  ்ଶ ൅ ܶ ൅ ܭܶ ൌ ቀܭ ൅

 

ଶቁ ܶ. 

 

In WPM, latency comes from the P/S and S/P conversion 

pair and filtering. The latency caused by filtering is ൫ܮ௤ െ

1ሻሺܮ െ 1ሻ ௦ܶ ൌ ൫ܮ௤ െ 1൯ሺܶ െ ௦ܶሻ , then the total latency is 

߬ௐ௉ெ ൌ ൫ܮ௤ െ 1൯ሺܶ െ ௦ܶሻ ൅ ܶ ൌ ܮ௤ܶ െ ൫ܮ௤ െ 1൯ ௦ܶ. 

 

Assuming the symbol duration T = 66.7μs, the latencies of 

OFDM for three different CP durations used in 3GPP LTE, of 

OFDM-OQAM for typical K values, and of WPM for various 

filters (Db1, Db4 and Db7 [7]) are shown in Table 3, Table 4 

and Table 5, respectively. Note that Table 5 shows the latency 

upper bounds by assuming ܮ ൌ ∞. It is clear that the latency 

from MCM for OFDM-OQAM and WPM is much higher than 

that for OFDM with CP. However, comparing to the latency 

requirement in IMT-Advanced (100ms for control plane and 

10ms for user plan), such an increase in latency should still be 

manageable, especially for OFDM-OQAM. 

 

V. EVALUATION OF TRANSMITTED SIGNAL 

Simulations are carried out for a system with 64 subcarriers. 

 

Assuming subcarrier spacing  ∆݂ ൌ 15݇ܪݖ or the symbol 

duration T = 66.7μs, the total bandwidth of the system is 960 

kHz. Let the number of samples per symbol, L, be 128 unless 

 

 

 

978-1-4577-1343-9/12/$26.00 ©2013 IEEE 

 

specified. Then, the sample duration is 0.52μs. QPSK symbol 

modulation is assumed unless specified. In OFDM, CP is not 

used. In OFDM-OQAM, the frequency sampling prototype 

filters (with three typical K values) that minimize the total 

structure based interference [3] are used. In WPM, two 

Daubechies wavelets, Db1 and Db7, are used.  

 

A. PAPR 

The PAPR of the lth transmitted symbol is defined as  

 

PAPR௟ ൌ  

݉ܽݔሺ௟ିଵሻ௅ஸ௡ஸ௟௅ିଵ|ݔሾ݊ሿ|

 

 

1

ܮ ∑ |ݔሾ݊ሿ|ଶ௟௅ିଵ௡ୀሺ௟ିଵሻ௅

 

 

(33)  

 

where ݔሾ݊ሿ is defined in (1). Fig.6 shows the complementary 

cumulative distribution function (CCDF) of PAPRs for 

different MCM signals when L = 64. The PAPR performance 

of OFDM-OQAM does not depend on the K value so that only 

the results of K = 3 is shown for simplicity. It is observed that 

the PAPR performance depends primarily on the number of 

subcarriers and is insensitive to the type of FBMC schemes 

considered in this paper. 

 

 

Figure 6. CCDF of the three FBMC systems. 

 

B. Power Spectral Density (PSD) 

Fig.7 shows the PSDs of the three types of FBMC signals. 

 

The OFDM-OQAM has the largest stopband attenuation 

because its prototype filter has small sidelobes. In addition, the 

stopband attenuation for K = 4 is about 20dB larger than that 

for K = 3. The WPM has the worst out-of-band leakage 

because its overall synthesis filters have large sidelobes. The 

WPM with Db7 has smaller out-of-band leakage than the 

WPM with Db1 since the former uses longer filter than that of 

latter. Note that in this simulation, the impact from nonlinear 

PA was not considered. With nonlinear PA, the out-of-band 

emission for all MCM schemes will increase and difference 

among them could be reduced. The investigation of the 

performance using nonlinear PA will be left as future work. 

 

VI. PERFORMANCE EVALUATION AT RECEIVER 

In this section, the simulation parameters are the same as 

 

those in Section V. 

 

A. Bit Error Rate (BER) 

Fig.8 shows the BER performance comparison using QPSK 

 

and 16QAM modulation in AWGN channel. The unit of the 

abscissa, Es/No, stands for the symbol signal-to-noise ratio. 

The three FBMC systems obtain almost the same BER 

 

performances for each modulation scheme and they are 

consistent with the theoretical results.  

 

 

Figure 7. PSDs of OFDM, OFDM-OQAM (K = 3 & 4), and WPM (Db1 & 

 

Db7). 

 

 

Figure 8. BER performance of OFDM, OFDM-OQAM and WPM. 

 

B. Sensitivity to Timing Offset (TO) and Carrier Frequency 

Offset (CFO) 

 

As OFDM and other multicarrier schemes, OFDM-OQAM 

is sensitive to TO and CFO. Table 6 shows the real part of the 

estimated input symbol for OFDM-OQAM. In Table 6, ݐ଴ is 

the TO in seconds, and 

 

߶൫݊, ଴݂, ሺ݉ െ ݇ሻ൯ ൌ ෍ ܲሺ݂ െ ∆݂ሻܲሺെ݂ െ ∆݂ሻ௙ ݁

௝ଶగ௙௡ (34)

 

represents the interference of the signal modulated at the mth 

subcarrier to that at the kth subcarrier due to CFO ଴݂  with 

 

∆݂ ൌ ௙బଶ ൅ ሺ݉ െ ݇ሻ

 

ଶ௅ and 

 

ݍሺݔሻ ൌ ෍ ܲ ൬݂ െ 12ܮ൰ ܲ ൬െ

1

 

2ܮ െ ݂൰ ݁

௝ଶగ௙௫

 

 

 

(35)

 

where ܲሺ݂ሻ is the Fourier transform of ݌ሺ݊ሻ. It is shown that 

both ICI and ISI are generated when there is TO or CFO. 

Since the stopband attenuation is large for OQAM-OFDM, the 

TO-generated-ICI only exists between adjacent subcarriers. 

This feature can be used in designing TO estimation. 

 

Fig.9 and Fig.10 display the BER performance degradation 

of the three FBMC systems due to TO and CFO, respectively. 

For OFDM, CP was not included so that all MCM schemes 

have the same bandwidth efficiency. Here we assume the 

channel is noise free so that the only distortion to the signal is 

from the ISI and ICI. For the case with CFO, pilots are 

 

4 5 6 7 8 9 10 11

10-3

 

10-2

 

10-1

 

100

 

PAPR

 

 

reference value (dB)

 

CC

DF

 

 

 

 

 

OFDM

WPM(db1)

OFDM-OQAM(K=3)

WPM(db7)

 

-1000 -500 0 500 1000

-100

 

-80

 

-60

 

-40

 

-20

 

0

 

20

 

f(kHz)

 

PS

D

 

(dB

)

 

 

 

 

 

WPM-Db7

WPM-Db1

OFDM

OFDM-OQAM(K=3)

OFDM-OQAM(K=4)

 

4 6 8 10 12 14 16 18 20

 

10-4

 

10-3

 

10-2

 

10-1

 

100

 

Es/N0(dB)

 

BE

R

 

 

 

 

 

Theory

OFDM

OFDM-OQAM(K=3)

WPM-Db1

WPM-Db7

 

Dashed line: 16QAM

Solid line:QPSK

 

 

 

978-1-4577-1343-9/12/$26.00 ©2013 IEEE 

 

inserted every 1ms for the receiver to correct the phase error 

caused by the CFO. From Fig.9 and Fig.10, we see that WPM 

(Db1) is least sensitive to TO and CFO, compared to others. 

Compared to OFDM, OFDM-OQAM is more sensitive to TO 

(or CFO) when TO (or CFO) is small but less sensitive to TO 

(or CFO) when TO (or CFO) is large. The break point depends 

on the constellation size. Besides, in asynchronous uplink or 

cognitive radio OFDM-OQAM systems are more robust than 

OFDM systems to misalignments among users because of its 

sufficient stopband attenuation. 

 

VII. CONCLUSION 

In this paper we reviewed three different MCM techniques, 

 

i.e., OFDM, OFDM-OQAM and WPM, and modeled them 

under the same FBMC framework. For OFDM-OQAM, we 

proposed an efficient implementation of the IFFT-based SFB 

and FFT-based AFB by using simple permutations to avoid 

complex multipliers. We compared these MCM techniques 

using multiple metrics. In general, OFDM-OQAM and WPM 

are more bandwidth efficient than OFDM since they don’t 

need CP. OFDM-OQAM has much lower out-of-band 

emission comparing to OFDM and WPM, which may make it 

more suitable to cognitive radio systems. On the other hand, 

for OFDM-OQAM the computational complexity is about 6 

times greater and the latency is 5 times longer than OFDM, 

but still in the manageable range for practical systems. For 

WPM, those numbers could be even larger. In terms of PAPR, 

the three MCM techniques with their nominal corresponding 

parameters have very similar characteristics. Regarding the 

timing and frequency offsets, WPM is least sensitive. OFDM-

OQAM is less sensitive to those offsets for large offset values, 

but more sensitive for small values. So, for OFDM-OQAM, to 

achieve better in-band performance, tighter time and 

frequency error control is needed. Timing and frequency error 

correction and the investigation into performance with a 

nonlinear PA is left as future work. 

 

REFERENCES 

[1] P. Siohan, C. Siclet and N. Lacaille, "Analysis and design of 

 

OFDM/OQAM systems based on filterbank theory," Signal Processing, 

IEEE Transactions on, vol.50, no.5, pp.1170-1183, May 2002. 

 

[2] A. Viholainen, T. Ihalainen, T. Hidalgo, M. Renfors and M. Bellanger. 

“Prototype filter design for filter bank based multicarrier transmission,” 

in Proc. of 17th European Signal Processing Conference, Aug. 2009. 

 

[3] A. Viholainen, M. Bellanger and M. Huchard, “Prototype filter and 

structure optimization,” Physical Layer for Dynamic Spectrum Access 

and Cognitive Radio, Jan. 2009. 

 

[4] B. Farhang-Boroujeny, "Tutorial: Filter Bank Multicarrier for Next 

Generation of Communication Systems," 2010 Wireless Symposium and 

Summer School at Virgina Tech, June 3, 2010. 

 

[5] A. Jamin and P. Mahonen, “Wavelet packet Modulation for Wireless 

Communications,” Wireless Communications & Mobile Computing 

Journal, John Wiley and Sons Ltd., vol. 5, no. 2, p. 123–137, Mar. 2005. 

 

[6] M. Gautier, C. Lereau, M. Arndt, and J. Lienard, “PAPR analysis in 

wavelet packet modulation,” in Proc. of 3rd International Symposium 

on Communications, Control and Signal Processing (ISCCSP), Mar. 

2008. 

 

[7] I. Daubechies, Ten Lectures on Wavelets, SIAM Publications, 1992. 

[8] A. Viholainen, J. Alhava, and M. Renfors, “Efficient implementation of 

 

complex modulated filter banks using cosine and sine modulated filter 

banks,” EURASIP Journal on Advances in Signal Processing, vol. 2006, 

Article ID 58564, 10 pages, doi: 10.1155/ASP/2006/58564, 2006. 

 

[9] B. Farhang-Boroujeny and C. H. G. Yuen, “Cosine modulated and offset 

QAM filter bank multicarrier techniques: A continuous-time prospect,” 

EURASIP Journal on Advances in Signal Processing, vol. 2010, Article 

ID 165654, 16 pages, doi: 10.1155/2010/165654, 2010. 

 

[10] M. K. Lakshmanan and H. Nikookar, “A Review of Wavelets for Digital 

Wireless Communication,” Wireless Personal Communications: An 

International Journal, vol.37 no.3-4, p. 387-420, May 2006. 

 

[11] H. S. Malvar, Signal Processing with Lapped Transforms, Artech House, 

Boston, Mass, USA, 1992. 

 

 

 

TABLE 6 ESTIMATED DATA AT THE RECEIVER 

 ݏҧመ௞,௚ூ  

 

CFO

: ଴݂ 

Hz 

 

෍ ෍

 

ۉ

ۈ

ۈ

ۈ

ۈ

ۇ

 

ܿ݋ݏ ቆߨ ଴݂ሺ݃ ൅ ݈ሻܮ ൅ ߨሺ݉ െ ݇ሻ ൬݃ െ ݈ ൅

1

2൰ቇ

 

כ ݏ௠,௟ூ ߶൫ሺ݃ െ ݈ሻܮ, ଴݂, ሺ݉ െ ݇ሻ൯ ൅

ܿ݋ݏ ൬ߨ ଴݂ ൬݃ ൅ ݈ ൅

 

1

2൰ ܮ ൅ ߨሺ݉ െ ݇ሻሺ݃ െ ݈ ൅ 1ሻ ൅

 

ߨ

2൰

 

כ ݏ௠,௟ொ ߶ ቆ൬݃ െ ݈ െ

1

2൰ ܮ, ଴݂, ሺ݉ െ ݇ሻቇ ی

 

ۋ

ۋ

ۋ

ۋ

ۊ

 

 

௟ୀି∞

 

ெିଵ

 

௠ୀ଴

 

TO:

݊଴ ൌ

ݐ଴ܮ/

 

ܶ ෍ ෍

 

ۉ

ۈۈ

ۈۈ

ۈ

ۇ

 

ܿ݋ݏ ቆሺ݉ െ ݇ሻ ߨ2 െ

ߨ

ܮ ሺ݉ ൅ ݇ሻ݊଴ ൅ ߨሺ݃ െ ݈ሻቇ

 

כ ݏ௠,௟ூ ݍ൫ሺ݃ െ ݈ሻܮ െ ݊଴൯ ൅

ܿ݋ݏ ቆ3ሺ݉ െ ݇ሻ ߨ2 െ

 

ߨ

ܮ ሺ݉ ൅ ݇ሻ݊଴ ൅ ߨሺ݃ െ ݈ሻቇ

 

כ ݏ௠,௟ொ ݍ ቆ൬݃ െ ݈ െ

1

2൰ ܮെ݊଴ቇ ی

 

ۋۋ

ۋۋ

ۋ

ۊ

 

 

௟ୀି∞

 

௞ାଵ

 

௠ୀ௞ିଵ

 

 

 

 

 

Figure 9. BER performance as function of TO. 

 

 

Figure 10. BER performance as function of CFO. 

 

0 50 100 150 200 250 300

0

 

0.05

 

0.1

 

0.15

 

0.2

 

0.25

 

0.3

 

0.35

 

Frequency Offset(Hz)

 

BE

R

 

 

 

 

 

OFDM-OQAM (K=3)

OFDM

WPM (Db1)

 

Dashed line: QPSK

Solid line:16QAM