# Cyclic Delay Transmission for Vector OFDM Systems

###### Abstract

Single antenna vector OFDM (V-OFDM) system has been proposed and investigated in the past. It contains the conventional OFDM and the single carrier frequency domain equalizer (SC-FDE) as two special cases and is flexible to choose any number of symbols in intersymbol interference (ISI) by choosing a proper vector size. In this paper, we develop cyclic delay diversity (CDD) transmission for V-OFDM when there are multiple transmit antennas (CDD-V-OFDM). Similar to CDD-OFDM systems, CDD-V-OFDM can also collect both spatial and multipath diversities. Since V-OFDM first converts a single input single output (SISO) ISI channel to a multi-input and multi-output (MIMO) ISI channel of order/length times less, where is the vector size, for a given bandwidth, the CDD-V-OFDM can accommodate times more transmit antennas than the CDD-OFDM does to collect all the spatial and multipath diversities. This property will specially benefit a massive MIMO system. We show that with the linear MMSE equalizer at each subcarrier, the CDD-V-OFDM achieves diversity order , where is the transmission rate, is the number of transmit antennas, and is the ISI channel length between each transmit and receive antenna pair. Simulations are presented to illustrate our theory.

## I Introduction

Orthogonal frequency division multiplexing (OFDM) systems have been well accepted in broadband wireless communications, such as LTE [1], WiMAX [2] and WiFi [2] systems. It is mainly because OFDM converts an ISI channel to multiple ISI free subchannels by using IDFT/DFT operations. When transmission data rates get higher and higher, signal bandwidths need to get wider and wider, which causes communication/ISI channel lengths longer and longer. A longer ISI channel forces the number of the subcarriers in an OFDM system larger in order to provide ISI free subchannels. A larger may cause more difficulties in OFDM implementations, such as high peak-to-average power ratio (PAPR), etc.

Single antenna V-OFDM is proposed in [3, 4], which first converts a SISO ISI channel into an equivalent MIMO channel with memory, but the MIMO channel memory length (or order) is reduced by times compared to that of the SISO ISI channel, and then converts the equivalent MIMO channel with memory to multiple MIMO subchannels without memory. Since the MIMO channel length is times less, the CP length added to the V-OFDM can be also reduced by times compared to that of the OFDM for the SISO ISI channel. The cost for V-OFDM is that, although the multiple MIMO subchannels are constant vector/matrix channels and do not have memory, in each MIMO subchannel, the information symbols in each signal vector are still ISI channel. In other words, V-OFDM does not convert an ISI channel completely to multiple ISI free subchannels as OFDM does, but converts an ISI channel to multiple sub-channels with a fixed and flexible number, , of ISI symbols in each subchannel and can be arbitrarily chosen. As the two extreme cases of , the conventional OFDM and single-carrier frequency domain equalizer (SC-FDE) correspond to the cases when and , respectively. With V-OFDM, when an ISI channel length increases as its bandwidth increases, the number of subcarriers may be fixed, while the vector size can be increased, which may be an alternative choice for a wide band system.

In wireless communications, an important technique to combat fading is to use multiple transmit antennas to collect spatial diversity [5, 6, 7]. For a broadband MIMO system, there are two kinds of diversities, namely spatial diversity and multipath diversity. To collect both spatial and multipath diversities, space-frequency/time coding has been proposed in the literature to code information symbols across not only antennas but also subcarriers, see for example [8, 9, 10, 11], which, however, has a high decoding complexity at the receiver. A much simpler technique to collect both spatial and multipath diversities is the cyclic delay diversity (CDD) technique [12, 13, 14, 15, 16, 17, 18, 19, 20, 21], where the other transmit antennas transmit cyclically delayed versions of the signal transmitted at the first transmit antenna in every OFDM block. With CDD-OFDM, to collect the full spatial and multipath diversities, the product of the number of transmit antennas with the ISI channel length cannot be more than the number of subcarriers. When transmit antenna number is large, the full spatial and multipath diversities may not be collected by using the CDD technique when the number of subcarriers and the channel bandwidth are fixed.

In this paper, we propose a CDD transmission for V-OFDM (CDD-V-OFDM) for multiple transmit antenna systems. Similar to the CDD-OFDM, the other transmit antennas transmit cyclically delayed versions of the V-OFDM signals the first transmit antenna transmits in every V-OFDM block. Since an ISI channel length can be equivalently reduced by times in V-OFDM, with our proposed CDD-V-OFDM, for a fixed channel bandwidth (or channel length) and a fixed IDFT/DFT size (or number of subcarriers), CDD-V-OFDM can accommodate times more transmit antennas than CDD-OFDM does, where full spatial and multipath diversities can be collected. This property may benefit a massive MIMO system, where a massive number of transmit antennas are used. Or, if the number of transmit antennas and channel bandwidth are fixed, the IDFT/DFT size can be reduced by times with the CDD-V-OFDM that still collects the full spatial and multipath diversities, compared to that of the CDD-OFDM, which will consequently reduce the PAPR by times.

As mentioned earlier, a V-OFDM [3, 4, 22, 23, 24, 25, 26] converts an ISI channel to multiple constant matrix/vector channels, each of which has information symbols together. When is not small, the maximum-likelihood (ML) decoding of such a constant matrix/vector subchannel may be complex. In this paper, we investigate the minimum mean squared error (MMSE) equalizer for these subchannels. Following the results obtained in [26], we show that with the linear MMSE equalizer at each subcarrier, the CDD-V-OFDM achieves diversity order , where is the transmission rate, is the number of transmit antennas, and is the ISI channel length between each transmit and receive antenna pair.

This paper is organized as follows. In Section \@slowromancapii@, we first review OFDM and V-OFDM systems. In Section \@slowromancapiii@, we derive a CDD-V-OFDM system, obtain an equivalent SISO ISI channel from the CDD-V-OFDM system for multiple transmit antennas, which also has an equivalent V-OFDM system. Then, we show the diversity order when linear MMSE equalizer is applied at each subchannel in the CDD-V-OFDM system. In Section IV, we present some simulation results to illustrate the analysis. In Section V, we conclude this paper.

Notations: Throughout the paper. denotes the Hermitian conjugate transpose of a matrix or a vector. denotes the transpose. and denote the modulo and the modulo operations. The bold face letters denote a vector or matrix. and denote the ceiling and the flooring operations, respectively.

## Ii Brief Review of OFDM, and V-OFDM

In this section, the conventional OFDM, and the V-OFDM system for single transmit antenna [3, 4] are briefly reviewed. The conventional OFDM system is shown in Fig. 1. is the frequency domain signal, is the number of the subcarriers in the OFDM system, and is the cyclic prefix length. After IDFT, the transmitted signal in time domain is

(1) |

and then the transmitted signal with CP is

(2) |

Assume that the ISI channel disperse has the following transfer function:

(3) |

and

h | ||||

(4) |

is the corresponding channel impulse response (CIR) in time domain. At the receiver, CP is removed first, and we get

(5) |

Then, transfer the received signal to frequency domain by DFT with size , we obtain

(6) |

Fig. 2 illustrates the block diagram of the V-OFDM system. Data block is transmitted for each V-OFDM block, which contains vectors with symbols in each vector. (If we keep the block length unchanged, the block can be divided into small vectors with low PAPR and save CP, here we use as the block length). Let

(7) |

where , and . V-OFDM does component-wise IDFT of size over the vectors, i.e.,

(8) |

we have the signal in time domain as

(9) |

Then, the transmitted symbol sequence after adding cyclic prefix (CP) in time domain can be written as

(10) |

where , and . After removing the CP at the receiver, we have the signal in time domain as

(11) |

where . With component-wise DFT in the similar way as (8), the signal will be transformed to frequency domain as

(12) |

where . The relationship between the transmitted and received signals in z domain can be expressed as follows. From [4], assume that and are the th polyphase components of the z-transforms of the transmitted and received signals, respectively. Then, the received signal can be written in the following MIMO form in domain

(13) |

where , , is the transform of the noise,

(14) |

and

(15) |

where

(16) |

is the th polyphase component of (II) and (II) is the th polyphase component of (3). Note that, for the convenience later, we use length for the th polyphase component in time domain as well but its number of non-zero components is at most . It is clear that (II) has the same form as the conventional OFDM system (II) for each , while the channel length is at most as shown in (II).

## Iii Proposed CDD-V-OFDM system

In this section, we first describe our proposed CDD-V-OFDM system and present some of its properties. We then present its performance analysis when the linear MMSE equalizer/receiver is used for every subchannel. Also, for convenience, in this section, we assume for some positive integer . A general case of can be similarly studied but the notations become more tedious.

### Iii-a Proposed System Description

Combining V-OFDM with CDD transmission can enhance the diversity order and accommodate more antennas in the system, especially in massive MIMO system as we shall see in this section. Our proposed CDD-V-OFDM system is shown in Fig. 3. In this section, the CDD-V-OFDM system and its equivalent vectorized MIMO ISI CIR are presented, where a single receive antenna is used for convenience.

Assume the original data sequence as , which is blocked into vectors with symbols in each vector:

(17) |

where and , . After the component-wise IDFT shown in (8), the signal in time domain is

(18) |

where , and , . There are transmit antennas in this system and let us consider the th transmit antenna. The cyclic shift with amount in the th transmit antenna leads the signal in (18) to

(19) |

where

(20) |

where . Then, the transmit signal after appending CP for the th transmit antenna is

(21) |

where and . Consider the original CIR from the th antenna as

(22) |

Then, the th polyphase of the received signal from the th antenna can be expressed as a linear convolution of and the original CIR from the th antenna as

(23) |

Referring to the V-OFDM, we need to transform the linear convolution between the CIR and the transmit signals to the cyclic convolution between the vectorized ISI channel and the vectorized OFDM signals. Then, we do the same vectorization operation for the ISI channel as the V-OFDM signal. Assume that , and , for , we have and is

(24) |

Then, the index of the signal in (III-A) is separated into two parts: one is the index of a symbol in a vector, i.e., , and the other is the index of the vector, i.e., . Substituting with , (III-A) is changed to

(25) |

Consider the first equation in (III-A), the transmit signal consists of vectors with symbols in each vector. Thus, (25) is changed to

(26) |

where and . Inside , , the region of CP is , which means, , for . After removing the CP at the receiver, the region is what we concern in the received signal. Here we divide the summation period of into two parts, and . Then, we have

(27) |

Because , for , the second item of (26) is changed to

(28) |

From (III-A) we know , for , then we have

(29) |

Substituting with , we have

(30) |

where . Then, the th polyphase component of the received signal can be expressed by the cyclic convolution of the vectorized OFDM signal and the vectorized ISI CIR as

(31) |

where denotes modulo operation and . For the cyclic shift in (18) and (19), we have . Then, (III-A) is rewritten as

(32) |

where, similar to (II), we have

(33) |

Substituting with , and following the similar process as (III-A)-(III-A), for , we obtain

(34) |

Since (III-A) is the signal from the th antenna, the overall received signal is the sum of the signals from all the antennas, which is

(35) |

where

(36) |

Equation (III-A) consists of polyphase components as . The equivalent SISO channel over all the polyphase components and all the antennas is

(37) |

Then, the th polyphase component of length in is

(38) |

Comparing (II) and (III-A), we can see that the equivalent ISI channel of the CDD-V-OFDM has the same form as that of V-OFDM for a SISO ISI channel, while in the CDD-V-OFDM system, the equivalent ISI channel is the sum of the cyclically shifted channels from different transmit antennas. Next, the vectorized ISI channel has the z-domain pseudocirculant matrix form as follows