# Bit Error Probability for MMSE Receiver in GFDM Systems

TL;DR: This letter considers the minimum mean-square error receiver for the generalized frequency division multiplexing system (GFDM) over frequency selective fading channels and derives an approximate probability density function for the signal-to-interference-plus-noise ratio.

Abstract: In this letter, we consider the minimum mean-square error receiver for the generalized frequency division multiplexing system (GFDM) over frequency selective fading channels. We derive an approximate probability density function for the signal-to-interference-plus-noise ratio. This expression allows us to obtain a new approximate, but rather accurate formulation for the bit error probability for a $\mathcal {M}$ -quadrature amplitude modulation scheme. Our results resort on the pivotal properties exhibited by eigenvalues of a circulant matrix. Since the entries of the channel matrix $\text {H}_{\text {ch}}$ are complex Gaussian distributed, and the eigenvalues are given as a weighted sum of its entries, the joint eigenvalue distribution is also Gaussian. Comparisons of the simulated and analytical results validate our formulation and allow a quick and efficient tool to compute the bit error rate for the GFDM system.

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##### Citations

22 citations

### Cites methods from "Bit Error Probability for MMSE Rece..."

...BIT ERROR PROBABILITY The bit error probability for the M-QAM and BPSK modulations can be found according to [32], [33] respectively as...

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7 citations

### Cites background from "Bit Error Probability for MMSE Rece..."

...From another point of view, although error probability with MMSE receiver is derived in [10], theoretical derivations are given according to an approximated distribution of signal-to-interference-plus-noise ratio (SINR) and the generality is not as clearly illustrated as that in [1], [2], and [7] through calculating NEF....

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5 citations

### Additional excerpts

...where with the help of (14) and the constraint of (15), the probability density function (PDF) pγ pγnq can be approximated by [21] pγpγnq « 1 Γpκq θκ p1` γnq ́1 ́κ exp ˆ...

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5 citations

4 citations

### Cites background from "Bit Error Probability for MMSE Rece..."

...Generalized Frequency Division Multiplexing (GFDM) is a promising modulation candidate for the 5G waveform due to its capability in addressing various major 5G requirements, especially low latency in a tactile internet scenario [1], [3], [4]....

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##### References

2,231 citations

### "Bit Error Probability for MMSE Rece..." refers background or methods in this paper

...In order to address this constraint, we consider a matrix H given as [7]...

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...It, being a circulant matrix, possesses the normalized eigenvectors given by [7] φ j = 1 √ N 1, ω j , ω(2)j , ....

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...The corresponding eigenvalues are given by [7]...

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942 citations

### "Bit Error Probability for MMSE Rece..." refers background in this paper

...As it was stated in [9], the BER for the lth bit error probability of M-QAM constellation can be expressed as...

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711 citations

### "Bit Error Probability for MMSE Rece..." refers background in this paper

...In this letter, we have employed an analytical approach to calculate the SINR for GFDM waveform using the MMSE receiver, considering the influence of frequency selective fading channels....

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...So, x = Ad, where A is the modulation matrix GFDM and the vector d represents the complex data symbols d = [d0 d1 · · · dN−1]T with variance σ 2d ....

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...Index Terms— GFDM, Gamma approximation, MMSE, SINR....

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...One of these technologies that is being proposed for low latency and high throughput is called Generalized Frequency Division Multiplexing (GFDM) [1]....

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...After the CP removal, the received vector can be written as, y = HchAd + ν, (1) where ν is the AWGN vector of length N with variance σ 2ν and Hch is a circulant Toeplitz matrix based on vector h given as [5] Hch = ⎡ ⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣ h1 0 · · · 0 hL · · · h2 h2 h1 · · · 0 0 · · · h3 ... . . . · · · ... hL hL−1 · · · · · · · · · · · · 0 0 hL · · · · · · · · · · · · 0 ... . . . · · · ... 0 0 hL · · · · · · h1 ⎤ ⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (2) The received vector y is distorted due to (i) self-interference coming from GFDM inherent non-orthogonality, and (i i) frequency selectivity introduced by the channel impulse response....

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277 citations

### "Bit Error Probability for MMSE Rece..." refers methods in this paper

...[6] have presented an expression for the SINR of the nth data symbol given as, γn = 1 MMSEn − 1 = 1 (IN + p N (HchA)†HchA)−1 nn − 1,...

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223 citations

### "Bit Error Probability for MMSE Rece..." refers background in this paper

...and is nothing but the normalized DFT matrix [8]....

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