Tag: Rayleigh

Symbol Error rate for QAM (16, 64, 256,.., M-QAM)

In May 2008, we derived the theoretical symbol error rate for a general M-QAM modulation (in  Embedded.com, DSPDesignLine.com and dsplog.com) under Additive White Gaussian Noise. While re-reading that post, felt that the article is nice and warrants a re-run, using OFDM as the underlying physical layer. This post discuss the derivation of symbol error rate for a general M-QAM modulation. The companion Matlab script compares the theoretical and the simulated symbol error rate for 16QAM, 64QAM and 256QAM over OFDM in AWGN channel.

Enjoy and HAPPY NEW YEAR 2012 !!!

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Transmit beamforming

In this post lets discuss a closed-loop transmit diversity scheme, where the transmitter has the knowledge of the channel. As there is a feedback path required from the receiver, to communicate the channel seen by the receiver to the transmitter, the scheme is called closed-loop transmit diversity scheme. Recall that the transmit diversity using Space Time Coding (Alamouti STBC) does not require the knowledge of the channel. In this post, we will restrict our discussion to a 2 transmit, 1 receive case. We will assume that the channel is a flat fading Rayleigh multipath channel and the modulation is BPSK.

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Derivation of BPSK BER in Rayleigh channel

This is a guest post by Jose Antonio Urigüen who is an Electrical and Electronic Engineer currently studying an MSc in Communications and Signal Processing at Imperial College in London. This guest post has been created due to his own curiosity when reviewing some concepts of BER for BPSK in Rayleigh channnel published in the dsplog.com

From the post on BER for BPSK in Rayleigh channnel, it was shown that, in the presence of channel , the effective bit energy to noise ratio is .

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MIMO with ML equalization

We have discussed quite a few receiver structures for a 2×2 MIMO channel namely,

(a) Zero Forcing (ZF) equalization

(b) Minimum Mean Square Error (MMSE) equalization

(c) Zero Forcing equalization with Successive Interference Cancellation (ZF-SIC)

(d) ZF-SIC with optimal ordering and

(e) MIMO with MMSE SIC and optimal ordering

From the above receiver structures, we saw that MMSE equalisation with optimally ordered Successive Interference Cancellation gave the best performance. In this post, we will discuss another receiver structure called Maximum Likelihood (ML) decoding which gives us an even better performance. We will assume that the channel is a flat fading Rayleigh multipath channel and the modulation is BPSK.

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MIMO with MMSE SIC and optimal ordering

This post attempts to build further on the MIMO equalization schemes which we have discussed –

(a) Minimum Mean Square Error (MMSE) equalization,

(b) Zero Forcing equalization with Successive Interference Cancellation (ZF-SIC) and

(c) ZF-SIC with optimal ordering.

We have learned that successive interference cancellation with optimal ordering improves the performance with Zero Forcing equalization. In this post, we extend the concept of successive interference cancellation to the MMSE equalization and simulate the performance. We will assume that the channel is a flat fading Rayleigh multipath channel and the modulation is BPSK.

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MIMO with ZF SIC and optimal ordering

In previous posts, we had discussed equalization of a 2×2 MIMO channel with Zero Forcing (ZF) equalization and later, Zero Forcing equalization with successive interference cancellation (ZF-SIC). In this post, we will explore a variant of ZF-SIC called Zero Forcing Successive Interference Cancellation with optimal ordering. We will assume that the channel is a flat fading Rayleigh multipath channel and the modulation is BPSK.

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MIMO with Zero Forcing Successive Interference Cancellation equalizer

The post on MIMO with Zero Forcing equalizer discussed a probable way of equalizing a 2×2 MIMO channel. The simulated results with the 2×2 MIMO system with zero forcing equalizer showed matching results as obtained in for a 1×1 system for BPSK modulation in Rayleigh channel. In this post, we will try to improve the bit error rate performance by trying out Successive Interference Cancellation (SIC). We will assume that the channel is a flat fading Rayleigh multipath channel and the modulation is BPSK.

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MIMO with MMSE equalizer

In a previous post, we had discussed a 2×2 MIMO transmission using BPSK modulation in Rayleigh channel with a Zero Forcing equalizer. The simulated results with the 2×2 MIMO system  with zero forcing equalizer showed matching results as obtained in for a 1×1 system for BPSK modulation in Rayleigh channel. In this post, we will discuss a different equalization approach called Minimum Mean Square Error (MMSE) equalization. We will assume that the channel is a flat fading Rayleigh multipath channel and the modulation is BPSK.

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MIMO with Zero Forcing equalizer

We had discussed three Single Input Multiple Output (SIMO also known as receive diversity) schemes – Selection combining, Equal Gain Combining, Maximal Ratio Combining and a Multiple Input Single Output (MISO, also known as transmit diversity) scheme – Alamouti 2×1 STBC. Let us now discuss the case where there a multiple transmit antennas and multiple receive antennas resulting in the formation of a Multiple Input Multiple Output (MIMO) channel. In this post, we will restrict our discussion to a 2 transmit 2 receive antenna case (resulting in a 2×2 MIMO channel). We will assume that the channel is a flat fading Rayleigh multipath channel and the modulation is BPSK.

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Alamouti STBC

In the recent past, we have discussed three receive diversity schemes – Selection combining, Equal Gain Combining and Maximal Ratio Combining. All the three approaches used the antenna array at the receiver to improve the demodulation performance, albeit with different levels of complexity. Time to move on to a transmit diversity scheme where the information is spread across multiple antennas at the transmitter. In this post, lets discuss a popular transmit diversity scheme called Alamouti Space Time Block Coding (STBC). For the discussion, we will assume that the channel is a flat fading Rayleigh multipath channel and the modulation is BPSK.

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Maximal Ratio Combining (MRC)

This is the third post in the series discussing receiver diversity in a wireless link. Receiver diversity is a form of space diversity, where there are multiple antennas at the receiver. The presence of receiver diversity poses an interesting problem – how do we use ‘effectively‘ the information from all the antennas to demodulate the data. In the previous posts, we discussed selection diversity and equal gain combining (EGC).

In this post, we will discuss Maximal Ratio Combining (MRC). For the discussion, we will assume that the channel is a flat fading Rayleigh multipath channel and the modulation is BPSK.
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Equal Gain Combining (EGC)

This is the second post in the series discussing receiver diversity in a wireless link. Receiver diversity is a form of space diversity, where there are multiple antennas at the receiver. The presence of receiver diversity poses an interesting problem – how do we use ‘effectively‘ the information from all the antennas to demodulate the data. In the previous post, we discussed selection diversity. In this post, we will discuss equal gain combining (EGC). For the discussion, we will assume that the channel is a flat fading Rayleigh multipath channel and the modulation is BPSK.

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Selection Diversity

This is the first post in the series discussing receiver diversity in a wireless link. Receiver diversity is a form of space diversity, where there are multiple antennas at the receiver. The presence of receiver diversity poses an interesting problem – how do we use ‘effectively‘ the information from all the antennas to demodulate the data. There are multiple ways to approach the problem. The three typical approaches to be discussed are – selection diversity, equal gain combining and maximal ratio combining. In this post we will discuss selection diversity. For the discussion, we will assume that the channel is a flat fading Rayleigh multipath channel and the modulation is BPSK.

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BER for BPSK in OFDM with Rayleigh multipath channel

Mr. Lealem Tamirat, in a comment on BER for BPSK in Rayleigh channel, wondered about the performance of an OFDM modulated system in a frequency selective Rayeligh fading channel. My response was that,

Though the total channel is a frequency selective channel, the channel experienced by each subcarrier in an OFDM system is a flat fading channel with each subcarrier experiencing independent Rayleigh fading.

So, assuming that the number of taps in the channel is lower than the cyclic prefix duration (which ensures that there is no inter symbol interference), the BER for BPSK with OFDM in a Rayleigh fading channel should be same as the result obtained for BER for BPSK in Rayleigh fading channel.

Let us try to define a quick simulation to confirm the claim.

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