MIMO

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This article is about MIMO in wireless communication. For other uses, see MIMO (disambiguation).
Understanding of SISO, SIMO, MISO and MIMO

In radio, multiple-input and multiple-output, or MIMO (pronounced as "my-moh" and/or "me-moh"), is the use of multiple antennas at both the transmitter and receiver to improve communication performance. Multiple antennas may be used to perform smart antenna functions such as spreading the total transmit power over the antennas to achieve an array gain that incrementally improves the spectral efficiency (more bits per second per hertz of bandwidth) and/or achieving a diversity gain that improves the link reliability (reduces fading). However, today the term “MIMO” usually refers to a method for multiplying the capacity of a radio link by exploiting multipath propagation.[1] This modern MIMO is an essential element of wireless communication standards such as IEEE 802.11n (Wi-Fi), IEEE 802.11ac (Wi-Fi), 4G, 3GPP Long Term Evolution, WiMAX and HSPA+. More recently, MIMO has been applied to power line communications (PLC) for 3-wire installations as part of standard ITU G.hn and specification HomePlug AV2.[2] [3]

History of MIMO[edit]

First concepts[edit]

The earliest ideas in this field go back to work by AR Kaye and DA George (1970), Branderburg and Wyner (1974)[4] and W. van Etten (1975, 1976). Jack Winters and Jack Salz at Bell Laboratories published several papers on beamforming-related applications in 1984 and 1986.[5]

Principle[edit]

The multi-user MIMO concept of space-division multiple access (SDMA) was proposed by Richard Roy and Björn Ottersten, researchers at ArrayComm, in 1991. Their US patent (No. 5515378 issued in 1996[6]) emphasizes "an array of receiving antennas at the base station" and "plurality of remote users".

Arogyaswami Paulraj and Thomas Kailath proposed the concept of spatial multiplexing (SM) using MIMO in 1993. Their US patent (No. 5,345,599 issued in 1994[7]) emphasized "wireless broadcast communications" applications and splitting a high-rate signal "into several low-rate signals".

In 1996, Greg Raleigh, Gerard J. Foschini, and Emre Telatar refined new approaches to MIMO technology, considering a configuration where multiple transmit antennas are co-located at one transmitter to improve the link throughput effectively.[8][9][10]

Bell Labs was the first to demonstrate a laboratory prototype of spatial multiplexing in 1998, where spatial multiplexing is a principal technology to improve the performance of MIMO communication systems.[11]

Wireless standards[edit]

In the commercial area, Iospan Wireless Inc. developed the first commercial system in 2001 that used MIMO with orthogonal frequency-division multiple access technology (MIMO-OFDMA). Iospan technology supported both diversity coding and spatial multiplexing. In 2005, Airgo Networks had developed an IEEE 802.11n precursor implementation based on their patents on MIMO-OFDM. Following that in 2006, several companies (including at least Broadcom, Intel, and Marvell) fielded a MIMO-OFDM solution based on a pre-standard for 802.11n Wi-Fi standard. Also in 2006, several companies (Beceem Communications, Samsung, Runcom Technologies, etc.) had developed MIMO-OFDMA based solutions for IEEE 802.16e WiMAX broadband mobile standard. All upcoming 4G systems will also employ MIMO technology. Several research groups have demonstrated over 1 Gbit/s prototypes.

Functions of MIMO[edit]

MIMO can be sub-divided into three main categories, precoding, spatial multiplexing or SM, and diversity coding.

Precoding is multi-stream beamforming, in the narrowest definition. In more general terms, it is considered to be all spatial processing that occurs at the transmitter. In (single-stream) beamforming, the same signal is emitted from each of the transmit antennas with appropriate phase and gain weighting such that the signal power is maximized at the receiver input. The benefits of beamforming are to increase the received signal gain - by making signals emitted from different antennas add up constructively - and to reduce the multipath fading effect. In line-of-sight propagation, beamforming results in a well-defined directional pattern. However, conventional beams are not a good analogy in cellular networks, which are mainly characterized by multipath propagation. When the receiver has multiple antennas, the transmit beamforming cannot simultaneously maximize the signal level at all of the receive antennas, and precoding with multiple streams is often beneficial. Note that precoding requires knowledge of channel state information (CSI) at the transmitter and the receiver.

Spatial multiplexing requires MIMO antenna configuration. In spatial multiplexing, a high-rate signal is split into multiple lower-rate streams and each stream is transmitted from a different transmit antenna in the same frequency channel. If these signals arrive at the receiver antenna array with sufficiently different spatial signatures and the receiver has accurate CSI, it can separate these streams into (almost) parallel channels. Spatial multiplexing is a very powerful technique for increasing channel capacity at higher signal-to-noise ratios (SNR). The maximum number of spatial streams is limited by the lesser of the number of antennas at the transmitter or receiver. Spatial multiplexing can be used without CSI at the transmitter, but can be combined with precoding if CSI is available. Spatial multiplexing can also be used for simultaneous transmission to multiple receivers, known as space-division multiple access or multi-user MIMO, in which case CSI is required at the transmitter.[12] The scheduling of receivers with different spatial signatures allows good separability.

Diversity Coding techniques are used when there is no channel knowledge at the transmitter. In diversity methods, a single stream (unlike multiple streams in spatial multiplexing) is transmitted, but the signal is coded using techniques called space-time coding. The signal is emitted from each of the transmit antennas with full or near orthogonal coding. Diversity coding exploits the independent fading in the multiple antenna links to enhance signal diversity. Because there is no channel knowledge, there is no beamforming or array gain from diversity coding. Diversity coding can be combined with spatial multiplexing when some channel knowledge is available at the transmitter.

Forms of MIMO[edit]

Example of an antenna for LTE with 2 ports antenna diversity

Multi-antenna types[edit]

Multi-antenna MIMO (or Single user MIMO) technology has been developed and implemented in some standards, e.g., 802.11n products.

  • SISO/SIMO/MISO are special cases of MIMO
    • Multiple-input and single-output (MISO) is a special case when the receiver has a single antenna.
    • Single-input and multiple-output (SIMO) is a special case when the transmitter has a single antenna.
    • Single-input single-output (SISO) is a conventional radio system where neither the transmitter nor receiver has multiple antenna.
  • Principal single-user MIMO techniques
  • Some limitations
    • The physical antenna spacing is selected to be large; multiple wavelengths at the base station. The antenna separation at the receiver is heavily space-constrained in handsets, though advanced antenna design and algorithm techniques are under discussion. Refer to: multi-user MIMO

Multi-user types[edit]

Main article: Multi-user MIMO

Recently, results of research on multi-user MIMO technology have been emerging. While full multi-user MIMO (or network MIMO) can have a higher potential, practically, the research on (partial) multi-user MIMO (or multi-user and multi-antenna MIMO) technology is more active.

  • Multi-user MIMO (MU-MIMO)
    • In recent 3GPP and WiMAX standards, MU-MIMO is being treated as one of the candidate technologies adoptable in the specification by a number of companies, including Samsung, Intel, Qualcomm, Ericsson, TI, Huawei, Philips, Alcatel-Lucent, and Freescale. For these and other firms active in the mobile hardware market, MU-MIMO is more feasible for low-complexity cell phones with a small number of reception antennas, whereas single-user SU-MIMO's higher per-user throughput is better suited to more complex user devices with more antennas.
    • PU2RC allows the network to allocate each antenna to a different user instead of allocating only a single user as in single-user MIMO scheduling. The network can transmit user data through a codebook-based spatial beam or a virtual antenna. Efficient user scheduling, such as pairing spatially-distinguishable users with codebook-based spatial beams, is additionally discussed for the simplification of wireless networks in terms of additional wireless resource requirements and complex protocol modification. Recently, PU2RC is included in the system description documentation (SDD) of IEEE 802.16m (WiMAX evolution to meet the ITU-R's IMT-Advance requirements).
    • Enhanced multiuser MIMO: 1) Employs advanced decoding techniques, 2) Employs advanced precoding techniques
    • SDMA represents either space-division multiple access or super-division multiple access where super emphasises that orthogonal division such as frequency and time division is not used but non-orthogonal approaches such as superposition coding are used.
  • Macrodiversity MIMO
    • A form of space diversity scheme which uses multiple transmit or receive base stations for communicating coherently with single or multiple users which are possibly distributed in the coverage area, in the same time and frequency resource.[13][14][15]
    • The transmitters are far apart in contrast to traditional microdiversity MIMO schemes such as single-user MIMO. In a multi-user macrodiversity MIMO scenario, users may also be far apart. Therefore, every constituent link in the virtual MIMO link has distinct average link SNR. This difference is mainly due to the different long-term channel impairments such as path loss and shadow fading which are experienced by different links.
    • Macrodiversity MIMO schemes pose unprecedented theoretical and practical challenges. Among many theoretical challenges, perhaps the most fundamental challenge is to understand how the different average link SNRs affect the overall system capacity and individual user performance in fading environments.[16]
  • MIMO Routing
    • Routing a cluster by a cluster in each hop, where the number of nodes in each cluster is larger or equal to one. MIMO routing is different from conventional (SISO) routing since conventional routing protocols route node-by-node in each hop.[17]
  • Massive MIMO is a technology where the number of terminals is much less than the number of base station (mobile station) antennas.[18] In a rich scattering environment, the full advantages of the massive MIMO system can be exploited using simple beamforming strategies such as maximum ratio transmission (MRT) or zero forcing (ZF). To achieve these benefits of massive MIMO, accurate CSI must be available perfectly. However, in practice, the channel between the transmitter and receiver is estimated from orthogonal pilot sequences which are limited by the coherence time of the channel. Most importantly, in a multicell setup, the reuse of pilot sequences of several co-channel cells will create pilot contamination. When there is pilot contamination, the performance of massive MIMO degrades quite drastically. To alleviate the effect of pilot contamination, the work of [19] proposes a simple pilot assignment and channel estimation method from limited training sequences.

Applications of MIMO[edit]

Spatial multiplexing techniques make the receivers very complex, and therefore they are typically combined with Orthogonal frequency-division multiplexing (OFDM) or with Orthogonal Frequency Division Multiple Access (OFDMA) modulation, where the problems created by a multi-path channel are handled efficiently. The IEEE 802.16e standard incorporates MIMO-OFDMA. The IEEE 802.11n standard, released in October 2009, recommends MIMO-OFDM.

MIMO is also planned to be used in Mobile radio telephone standards such as recent 3GPP and 3GPP2. In 3GPP, High-Speed Packet Access plus (HSPA+) and Long Term Evolution (LTE) standards take MIMO into account. Moreover, to fully support cellular environments, MIMO research consortia including IST-MASCOT propose to develop advanced MIMO techniques, e.g., multi-user MIMO (MU-MIMO).

MIMO technology can be used in non-wireless communications systems. One example is the home networking standard ITU-T G.9963, which defines a powerline communications system that uses MIMO techniques to transmit multiple signals over multiple AC wires (phase, neutral and ground).[2]

Mathematical description[edit]

MIMO channel model

In MIMO systems, a transmitter sends multiple streams by multiple transmit antennas. The transmit streams go through a matrix channel which consists of all \scriptstyle N_t N_r paths between the \scriptstyle N_t transmit antennas at the transmitter and \scriptstyle N_r receive antennas at the receiver. Then, the receiver gets the received signal vectors by the multiple receive antennas and decodes the received signal vectors into the original information. A narrowband flat fading MIMO system is modelled as

\mathbf{y} = \mathbf{H}\mathbf{x} + \mathbf{n}

where \scriptstyle\mathbf{y} and \scriptstyle\mathbf{x} are the receive and transmit vectors, respectively, and \scriptstyle\mathbf{H} and \scriptstyle\mathbf{n} are the channel matrix and the noise vector, respectively.

Referring to information theory, the ergodic channel capacity of MIMO systems where both the transmitter and the receiver have perfect instantaneous channel state information is[20]

C_\mathrm{perfect-CSI} = E\left[\max_{\mathbf{Q}; \, \mbox{tr}(\mathbf{Q}) \leq 1} \log_2 \det\left(\mathbf{I} + \rho \mathbf{H}\mathbf{Q}\mathbf{H}^{H}\right)\right] = E\left[\log_2 \det\left(\mathbf{I} + \rho \mathbf{D}\mathbf{S} \mathbf{D} \right)\right]

where \scriptstyle ()^H denotes Hermitian transpose and \scriptstyle\rho is the ratio between transmit power and noise power (i.e., transmit SNR). The optimal signal covariance \scriptstyle \mathbf{Q}=\mathbf{VSV}^H is achieved through singular value decomposition of the channel matrix \scriptstyle\mathbf{UDV}^H \,=\, \mathbf{H} and an optimal diagonal power allocation matrix \scriptstyle \mathbf{S}=\textrm{diag}(s_1,\ldots,s_{\min(N_t, N_r)},0,\ldots,0). The optimal power allocation is achieved through waterfilling,[21] that is

s_i = \left(\mu - \frac{1}{\rho d_i^2} \right)^+, \quad \textrm{for} \,\, i=1,\ldots,\min(N_t, N_r),

where \scriptstyle d_1,\ldots,d_{\min(N_t, N_r)} are the diagonal elements of \scriptstyle \mathbf{D}, \scriptstyle (\cdot)^+ is zero if its argument is negative, and \mu is selected such that \scriptstyle s_1+\ldots+s_{\min(N_t, N_r)}=N_t.

If the transmitter has only statistical channel state information, then the ergodic channel capacity will decrease as the signal covariance \scriptstyle \mathbf{Q} can only be optimized in terms of the average mutual information as[20]

C_\mathrm{statistical-CSI} = \max_{\mathbf{Q}} E\left[\log_2 \det\left(\mathbf{I} + \rho \mathbf{H}\mathbf{Q}\mathbf{H}^{H}\right)\right].

The spatial correlation of the channel has a strong impact on the ergodic channel capacity with statistical information.

If the transmitter has no channel state information it can select the signal covariance \scriptstyle \mathbf{Q} to maximize channel capacity under worst-case statistics, which means \scriptstyle \mathbf{Q}=1/N_t \mathbf{I} and accordingly

C_\mathrm{no-CSI} = E\left[\log_2 \det\left(\mathbf{I} + \frac{\rho}{N_t}\mathbf{H}\mathbf{H}^{H}\right)\right].

Depending on the statistical properties of the channel, the ergodic capacity is no greater than \scriptstyle\min(N_t, N_r) times larger than that of a SISO system.

MIMO testing[edit]

MIMO signal testing focuses first on the transmitter/receiver system. The random phases of the sub-carrier signals can produce instantaneous power levels that cause the amplifier to compress, momentarily causing distortion and ultimately symbol errors. Signals with a high PAR (peak-to-average ratio) can cause amplifiers to compress unpredictably during transmission. OFDM signals are very dynamic and compression problems can be hard to detect because of their noise-like nature.[22]

Knowing the quality of the signal channel is also critical. A channel emulator can simulate how a device performs at the cell edge, can add noise or can simulate what the channel looks like at speed. To fully qualify the performance of a receiver, a calibrated transmitter, such as a vector signal generator (VSG), and channel emulator can be used to test the receiver under a variety of different conditions. Conversely, the transmitter's performance under a number of different conditions can be verified using a channel emulator and a calibrated receiver, such as a vector signal analyzer (VSA).

Understanding the channel allows for manipulation of the phase and amplitude of each transmitter in order to form a beam. To correctly form a beam, the transmitter needs to understand the characteristics of the channel. This process is called channel sounding or channel estimation. A known signal is sent to the mobile device that enables it to build a picture of the channel environment. The mobile device sends back the channel characteristics to the transmitter. The transmitter can then apply the correct phase and amplitude adjustments to form a beam directed at the mobile device. This is called a closed-loop MIMO system. For beamforming, it is required to adjust the phases and amplitude of each transmitter. In a beamformer optimized for spatial diversity or spatial multiplexing, each antenna element simultaneously transmits a weighted combination of two data symbols.[23]

MIMO literature[edit]

Principal researches[edit]

Papers by Gerard J. Foschini and Michael J. Gans,[24] Foschini[25] and Emre Telatar[10] have shown that the channel capacity (a theoretical upper bound on system throughput) for a MIMO system is increased as the number of antennas is increased, proportional to the smaller of the number of transmit antennas and the number of receive antennas. This is known as the multiplexing gain and this basic finding in information theory is what led to a spurt of research in this area. Despite the simple propagation models used in the aforementioned seminal works, the multiplexing gain is a fundamental property that can be proved under almost any physical channel propagation model and with practical hardware that is prone to transceiver impairments.[26]

A textbook by A. Paulraj, R. Nabar and D. Gore has published an introduction to this area.[27] There are many other principal textbooks available as well.[28][29][30] Mobile Experts has published a research report which predicts the use of MIMO technology in 500 million PCs, tablets, and smartphones by 2016.

Diversity-multiplexing tradeoff (DMT)[edit]

There exists a fundamental tradeoff between transmit diversity and spatial multiplexing gains in a MIMO system (Zheng and Tse, 2003).[31] In particular, achieving high spatial multiplexing gains is of profound importance in modern wireless systems.[32]

Other applications[edit]

Given the nature of MIMO, it is not limited to wireless communication. It can be used for wire line communication as well. For example, a new type of DSL technology (gigabit DSL) has been proposed based on binder MIMO channels.

Sampling theory in MIMO systems[edit]

An important question which attracts the attention of engineers and mathematicians is how to use the multi-output signals at the receiver to recover the multi-input signals at the transmitter. In Shang, Sun and Zhou (2007), sufficient and necessary conditions are established to guarantee the complete recovery of the multi-input signals.[33]

See also[edit]

References[edit]

  1. ^ Lipfert, Hermann (August 2007). MIMO OFDM Space Time Coding – Spatial Multiplexing, Increasing Performance and Spectral Efficiency in Wireless Systems, Part I Technical Basis (Technical report). Institut für Rundfunktechnik. 
  2. ^ a b Berger, Lars T.; Schwager, Andreas; Schneider, Daniel M. (February 2014). MIMO Power Line Communications: Narrow and Broadband Standards, EMC, and Advanced Processing. Devices, Circuits, and Systems. CRC Press. ISBN 9781466557529. 
  3. ^ HomePlug AV2 Technology (Technical report). HomePlug Powerline Alliance, Inc. 2013. 
  4. ^ Brandenburg, LH; Wyner, AD (May–June 1974). "Capacity of the Gaussian Channel with Memory: The Multivariate Case". Syst. Tech. J. (Bell) 53 (5): 745–78. doi:10.1002/j.1538-7305.1974.tb02768.x. 
  5. ^ Salz, J (July–August 1985). "Digital transmission over cross-coupled linear channels". Technical Journal (AT&T) 64 (6): 1147–59. doi:10.1002/j.1538-7305.1985.tb00269.x. 
  6. ^ US 5515378, "Spatial division multiple access wireless communication systems" 
  7. ^ US 5345599, "Increasing capacity in wireless broadcast systems using distributed transmission/directional reception (DTDR)" 
  8. ^ Raleigh, Gregory G; Cioffi, John M (March 1998). "Spatio-temporal coding for wireless communication". Transactions on Communications (IEEE) 46 (3): 357–66. doi:10.1109/26.662641. 
  9. ^ Foschini, GJ (Autumn 1996). "Layered space–time architecture for wireless communication in a fading environment when using multiple antennas". Labs Syst. Tech. J. (Bell) 1: 41–59. 
  10. ^ a b Telatar, Emre (1999). "Capacity of Multi-antenna Gaussian Channels". European Transactions on Telecommunications 10 (6): 585–95. doi:10.1002/ett.4460100604. 
  11. ^ Golden, GD; Foschini, GJ; Valenzuela, RA; Wolniansky, PW (Jan 1999). "Detection algorithm and initial laboratory results using V-BLAST space–time communication architecture". Electron. Lett. 35: 14–16. doi:10.1049/el:19990058. 
  12. ^ D. Gesbert, M. Kountouris, R. W. Heath, Jr., C.-B. Chae, and T. Sälzer (Oct 2007). "Shifting the MIMO Paradigm: From Single User to Multiuser Communications". IEEE Signal Processing Magazine 24 (5): 36–46. doi:10.1109/msp.2007.904815. 
  13. ^ Karakayali, M.K.; Foschini, G.J.; Valenzuela, R.A. (2006). "Advances in smart antennas - Network coordination for spectrally efficient communications in cellular systems". IEEE Wireless Communications 13 (4): 56–61. doi:10.1109/MWC.2006.1678166. 
  14. ^ Gesbert, David; Hanly, Stephen; Huang, Howard; Shamai Shitz, Shlomo; Simeone, Osvaldo; Yu, Wei (2010). "Multi-Cell MIMO Cooperative Networks: A New Look at Interference". IEEE Journal on Selected Areas in Communications 28 (9): 1380–1408. doi:10.1109/JSAC.2010.101202. 
  15. ^ Björnson, Emil (2013). "Optimal Resource Allocation in Coordinated Multi-Cell Systems". Foundations and Trends® in Communications and Information Theory 9 (2–3): 113–381. doi:10.1561/0100000069. 
  16. ^ Basnayaka, Dushyantha A.; Smith, Peter J.; Martin, Phillipa A. (2013). "Performance Analysis of Macrodiversity MIMO Systems with MMSE and ZF Receivers in Flat Rayleigh Fading". IEEE Transactions on Wireless Communications 12 (5): 2240–2251. doi:10.1109/TWC.2013.032113.120798. 
  17. ^ S. Cui, A. J. Goldsmith, and A. Bahai (August 2004). "Energy-efficiency of MIMO and Cooperative MIMO in Sensor Networks". IEEE J. Select. Areas of Commun. 22 (6): 1089–1098. doi:10.1109/JSAC.2004.830916. 
  18. ^ T. L. Marzetta, Noncooperative cellular wireless with unlimited numbers of base station antennas, IEEE Trans. Wireless Commun., vol. 9, no. 11, pp. 3590 – 3600, Nov. 2010.
  19. ^ T. E. Bogale and L. B. Le, Pilot optimization and channel estimation for multiuser massive MIMO systems in Proc. IEEE Conference on Information Sciences and Systems (CISS), Princeton, USA, Mar. 2014.
  20. ^ a b Love, David; Heath, Robert; n. Lau, Vincent; Gesbert, David; Rao, Bhaskar; Andrews, Matthew (2008). "An overview of limited feedback in wireless communication systems". IEEE Journal on Selected Areas in Communications 26 (8): 1341–1365. doi:10.1109/JSAC.2008.081002. 
  21. ^ D. Tse and P. Viswanath, Fundamentals of Wireless Communication, Cambridge University Press, 2005.
  22. ^ Stefan Schindler, Heinz Mellein, "Assessing a MIMO Channel", Rohde & Schwarz, pg. 11.
  23. ^ Agilent [1], Agilent MIMO Channel Modeling and Emulation Test Challenges, pg. 10, January 22, 2010, accessed September 16, 2011.
  24. ^ Gerard J. Foschini and Michael. J. Gans (January 1998). "On limits of wireless communications in a fading environment when using multiple antennas". Wireless Personal Communications 6 (3): 311–335. doi:10.1023/A:1008889222784. 
  25. ^ Gerard J. Foschini (Autumn 1996). "Layered space-time architecture for wireless communications in a fading environment when using multi-element antennas". Bell Labs Technical Journal 1 (2): 41–59. doi:10.1002/bltj.2015. 
  26. ^ Emil Björnson, Per Zetterberg, Mats Bengtsson, Björn Ottersten; Zetterberg; Bengtsson; Ottersten (January 2013). "Capacity Limits and Multiplexing Gains of MIMO Channels with Transceiver Impairments". IEEE Communications Letters 17 (1): 91–94. arXiv:1209.4093. Bibcode:2012arXiv1209.4093B. doi:10.1109/LCOMM.2012.112012.122003. 
  27. ^ A. Paulraj, R. Nabar and D. Gore (2003). "Introduction to Space-time Communications". Cambridge University Press. 
  28. ^ David Tse, Pramod Viswanath (2005). "Fundamentals of Wireless Communication". Cambridge. 
  29. ^ Claude Oestges, Bruno Clerckx (2007). "MIMO Wireless Communications: From Real-world Propagation to Space-time Code Design". Academic Press. 
  30. ^ Ezio Biglieri, Robert Calderbank, Anthony Constantinides, Andrea Goldsmith, Arogyaswami Paulraj, H. Vincent Poor (2010). "MIMO Wireless Communications". Cambridge University Press. 
  31. ^ L. Zheng and D. N. C. Tse (May 2003). "Diversity and multiplexing: A fundamental tradeoff in multiple-antenna channels". IEEE Trans. Inf. Th. 49 (5): 1073–1096. doi:10.1109/TIT.2003.810646. 
  32. ^ A. Lozano and N. Jindal (2010). "Transmit diversity vs. spatial multiplexing in modern MIMO systems". IEEE Trans. Wireless Commun. 9 (1): 186–197. doi:10.1109/TWC.2010.01.081381. 
  33. ^ Z. Shang, W. Sun and X. Zhou (January 2007). "Vector sampling expansions in shift invariant subspaces". Journal of Mathematical Analysis and Applications 325 (2): 898–919. doi:10.1016/j.jmaa.2006.02.033. 

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