# Selecting FFT Word Length for an OFDM Receiver That Supports Undersampling

## Abstract

**:**

## 1. Introduction

## 2. Background

#### 2.1. Proposed Undersampling Method on OFDM Receiver Side

_{2}q bits are mapped to q-QAM constellation symbols X

_{k}(0≤k<N) that form the parallel input to an N-point IFFT. The output of this IFFT are the symbols x

_{n}(0≤n<N) that are serially transmitted over the channel. Pilot symbols with known value are placed on reserved subcarriers for channel estimation and equalization. A cyclic prefix is also appended to avoid Inter-Symbol Interference (ISI). Digital/Analog Conversion (DAC) is required for the transmission of the resulting symbols using an appropriate pulse shaping method [12,13]. In wired OFDM transceivers the channel noise is assumed to be Additive White Gaussian Noise (AWGN) and the y

_{n}symbols received are y

_{n}=x

_{n}+z

_{n}, (z

_{n}is the noise with variance ${\sigma}_{n}^{2}$). A different model is used for wireless channels that takes into consideration the reflections, interference, Rayleigh fading, etc. In optical communications the fiber channels are affected by several sources of distortion including Kerr non-linearity, chromatic dispersion, optical filtering, double Rayleigh scattering, shot and thermal noise and especially Amplified Spontaneous Emission (ASE) [12]. The y

_{n}symbols at the output of the receiver ADC, form the input of an FFT. The FFT output symbols Y

_{k}(0≤k<N) are mapped to the closest QAM symbol and then QAM demodulation is performed (e.g., using hard or preferably soft decision demodulators). Forward Error Correction (FEC) decoding (Viterbi, Turbo codes, etc) exploits the available parity bits in order to correct as many errors as possible on the receiver side.

^{2}+D

^{3}and 1+D+D

^{2}respectively, where the D

^{p}denotes a delay of p clock periods. In order to apply the proposed undersampling method, the Interleaver should generate q-QAM symbols derived from parity or data bits only. Thus, a pair of small buffers at the FEC encoder output can store temporarily log

_{2}(q) bits from the systematic and the parity output of the RSC encoder before they are mapped to the corresponding q-QAM symbol. Most of the q-QAM symbols derived from sparse data bits will have a common value X

_{c}. However, several parity q-QAM symbols derived from parity bits are likely to have identical values because the parity output of the employed RSC encoder described above remains ‘0’ until a first data ‘1’ appears. Then, the 7-bit pattern “0111010” is repeated as long as the data input remains ‘0’. A different 7-bit pattern is generated when another ‘1’ appears at the input and this is also repeated until a third ‘1’ appears. The distance between each ‘1’ at the FEC encoder input is expected to be high due to the sparseness in the input. Consequently, several identical consecutive 7-bit patterns are expected to appear at the parity output of the FEC encoder. These identical parity patterns can be treated as if they were X

_{c}symbols as will be explained below.

_{n}symbols since they can be substituted by others, that have already been received. The adopted symbol arrangement at the IFFT input, makes trivial several IFFT/FFT operations that can be omitted in order to achieve lower power and faster operation. This is similar to output pruning described in [14].

_{n}symbols at the FFT input can be replaced by their counterparts at distance N/2: y

_{n+N/2}. It is obvious that no error would occur only if all the data q-QAM symbols placed the odd positions were equal to X

_{c}, e.g., if the data input is constantly zero. An error occurs if some of the data q-QAM symbols at the odd positions are not trivial (not equal to X

_{c}). To reduce the probability of errors, the number of samples R that can be substituted at the input of the receiver FFT can be lower than N/4, e.g., N/8 or N/16 [8].

_{n}can be expressed as:

_{n}=x

_{N/2-n}(n<N/4) if the following three conditions hold: a) ${X}_{k}={X}_{\frac{N}{2}-k}$(with odd k and k≤N/4), b)${X}_{N-k}={X}_{\frac{N}{2}+k}$(with odd k and k≤N/4) and c) ${X}_{k}={X}_{N-k}$ with even k and 0≤k<N/2. In a similar way, it can be shown that x

_{N/2+n}=x

_{N-n}(n<N/4). Consequently, the y

_{n}samples with odd n≤N/4 can be substituted by the samples ${y}_{\frac{N}{2}-n}$ and the ${y}_{\frac{N}{2}+n}$ samples can be substituted by ${y}_{N-n}$.

_{c}). If the RSC FEC encoder described earlier is employed then, the repeated 7-bit parity pattern can be padded with one more bit. The Most Significant 4-bits (Parity MSB) and the Least Significant 4-bits (Parity LSB) of the padded 8-bit parity patterns are placed in Figure 1 in appropriate positions in order to apply the proposed undersampling scheme with the lower possible error. In [9], the proposed undersampling method is extended to wireless OFDM systems with STBC encoding and several other IFFT input packet structures like the one shown in Figure 1 are described.

#### 2.2. Review of Quantization and Round-off Error Estimation Methods

_{w}depends on the window function (m values between 3 and 10 are tested in [17]). This limit is compared in [17] against older stochastic approximations presented in [18] ($NSR={2}^{-2b}m/6$) and [19] ($NSR=2{m}^{2}{2}^{-2b}$).

_{ADC}-bit resolution ADC with reference voltage V

_{ref}, then $\mathit{\Delta}={V}_{ref}/{2}^{{r}_{ADC}}$ (see Figure 2). For normalization reasons we assume V

_{ref}=1. The error caused by the quantization process is between $-\mathit{\Delta}/2$. and $+\mathit{\Delta}/2$. The error probability is assumed to be uniform ($1/\mathit{\Delta}$) in these limits. The variance of the error is then estimated as: ${\sigma}_{QE}^{2}={\mathit{\Delta}}^{2}/12$. The RE error caused by the use of finite word-length in an N-point FFT is also viewed as QE in [10]. The quantization noise power P

_{QE}in all real and imaginary parts of the DFT outputs as defined in Equation (1) is estimated in [10] as:

_{QE}noise is reduced as indicated by the relation (8) below [10]. The upper limit of the relation (8) corresponds to the classic FFT implementation by Cooley and Turkey [20].

_{i}(0≤i≤d) is the corresponding additive N×1 additive noise vector of ${w}_{Fi}$ (the equivalent twiddle factor matrix at the i-th stage of DIF FFT: ${w}_{{F}_{i}}={w}_{{T}_{d-i-1}}$) with variance ${\sigma}_{c}^{2}$. The total quantization noise power P

_{nt}of the DIT FFT algorithm is:

^{p-1}-1 and ${\sigma}_{h}^{2}={2}^{-2b-4}$ if the operations are performed with b+1 precision. The parameter ${\sigma}_{cs}^{2}$ is the variance of the discretization error caused by the multiplication with the cos/sin coefficients. These theoretical models are statistically checked in [24] using DIT FFT with two rounding methods.

_{f}is a fraction or a multiple of UE depending on the application specifications.

## 3. Proposed UE, RE Error Model

_{UE}) is compared to the QE error (P

_{QE}) in order to find an acceptable word length for the FFT.

_{i}symbol appears at the even input i+2 of the four-point FFT. The error in this case is ${e}_{logN-1}^{\left(i+2\right)}=\left[\epsilon {w}^{0}\epsilon {w}^{{2}^{logN-2}}\epsilon {w}^{0}\epsilon {w}^{{2}^{logN-2}}\right]$. In the third case the error appears at the odd input i+1: ${e}_{logN-1}^{\left(i+1\right)}=\left[\epsilon {w}^{0}-\epsilon {w}^{0}\epsilon {w}^{0}-\epsilon {w}^{0}\right]$. In the last case at the bottom of Figure 2, the error at the odd input i+3 is ${e}_{logN-1}^{\left(i+3\right)}=\left[\epsilon {w}^{0}{w}^{0}-\epsilon {w}^{0}{w}^{{2}^{logN-3}}-\epsilon {w}^{0}{w}^{0}\epsilon {w}^{0}{w}^{{2}^{logN-3}}\right]$. In the previous expressions of e

_{logN-1}the twiddle factors ${w}_{N}^{j}$ are used without the index N for simplicity. The differences in the four e

_{logN-1}expressions are owed to the different twiddle factors that multiply the error ε as it propagates through different paths. Figure 3 shows how the e

_{logN-I}error of stage logN-p propagates to the next butterfly stage logN-(p-1). The arrows in Figure 3 are buses and Figure 3a shows the propagation of the error from the top butterfly input, while Figure 3b shows the propagation of the error from the bottom input. In the first case ${e}_{logN-\left(p-1\right)}=\left[{e}_{logN-p}{e}_{logN-p}\right]$ while in the latter case ${e}_{logN-\left(p-1\right)}=\left[w.\ast {e}_{logN-p}\u2013w.\ast {e}_{logN-p}\right]$. The symbol “.*” implies multiplication of the corresponding elements of the vectors and w is the vector of all the twiddles of a specific butterfly stage. If multiple errors exist in the FFT inputs, their effect is added at the FFT output. For example, if the errors ε

_{3}and ε

_{7}occur in the bit reversed 16-point DIT FFT inputs y

_{3}and y

_{7}(corresponding to Y

_{12}and Y

_{14}outputs), the individual output errors (${e}_{0}^{{y}_{3}}$ and ${e}_{0}^{{y}_{7}}$, respectively) would be:

_{3}and ε

_{7}occur is ${e}_{0}^{{y}_{3},{y}_{7}}={e}_{0}^{{y}_{3}}+{e}_{0}^{{y}_{7}}$. Of course, the same error propagation model holds for the IFFT on the transmitter side. Moreover, the estimation of the error ε is easier on the transmitter side since, the IFFT inputs are QAM symbols with integer values. The 16-QAM constellation shown in Figure 4 is used as a case study, and we assume that X

_{c}= ”1111”, i.e., the trivial input is ‘1’. We can see that there are: a) N

_{(2)}= 4 neighboring QAM symbols (X

_{(2)}) that differ by 2 in the real or imaginary direction from X

_{c}(the continuous arrows), b) N

_{(2,2)}= 4 symbols (X

_{(2,2)}) that differ by 2 in each direction from X

_{c}(dashed arrows with big dashes), c) N

_{(2,4)}= 4 symbols (X

_{(2,4)}) that differ by 2 in one direction and by 4 in the other from X

_{c}(dotted arrows), d) N

_{(4)}= 2 symbols (X

_{(4)}) that differ by 4 in either the imaginary or real direction from X

_{c}(dashed arrows with small dashes), and e) a single (N

_{(4,4)}= 1) symbol (X

_{(4,4)}) that differs from X

_{c}by 4 in each direction. X

_{(2)}symbols correspond to 4 input bits with one ‘0’ while X

_{(2,2)}and X

_{(4)}symbols are derived from 4 input bits with 2 zeros. Finally, the X

_{(2,4)}symbols correspond to 4 input bits with 3 zeros and the QAM symbol (-3,-3) is derived by “0000”. The corresponding probability of each symbol is p

_{c}, p

_{(2)}, p

_{(2,2)}, p

_{(4)}, p

_{(2,4)}, p

_{(4,4)}. The order of these probabilities is p

_{c}> p

_{(2)}> p

_{(2,2)}= p

_{(4)}> p

_{(2,4)}> p

_{(4,4)}due to data sparseness.

_{c}= (1,1), the minimum error in 16-QAM modulation is ${\epsilon}_{min}=\sqrt{{\left(1-1\right)}^{2}+{\left(1-\left(-1\right)\right)}^{2}}=2={\epsilon}_{\left(2\right)}$. The rest of the errors are: ${\epsilon}_{\left(2,2\right)}=\sqrt{2\ast {2}^{2}}=2\sqrt{2}$, ${\epsilon}_{\left(4\right)}=\sqrt{{4}^{2}}=4$, ${\epsilon}_{\left(2,4\right)}=\sqrt{{2}^{2}+{4}^{2}}=2\sqrt{5}$, ${\epsilon}_{\left(4,4\right)}=\sqrt{2\ast {4}^{2}}=4\sqrt{2}$. The expected value of ε will be:

_{c}are taken into consideration and they have equal probability. In general, E[ε] depends on the sparseness level s<1 of the input and the QAM modulation. The sparseness level s means that a fraction s of the input data bits is non-trivial. If all the non-trivial symbols have equal probability to appear, then p

_{(k)}=s/(N-1). In this case, Equation (15) can be rewritten as:

_{x}is the decision region (Voronoi) of X. In the 16QAM modulation examined earlier, if one of the constellation bits is inverted, the effect on the BER of this error is 1/4 of the effect on the SER. It can be stated that SER represents the worst case effect of the error to the OFDM system. This is also confirmed by various simulations performed in the Appendix C of [12] where several modulations schemes are tested.

_{2}N bits. For example, in Figure 2, the dashed line that reaches output Y

_{12}shows a potential path that the error has followed from input y

_{6}. The error propagation path in this case can be denoted by “0110” and the initial error ε can be multiplied in each branch by 1, a twiddle factor w or –w. Table 1, lists for example, all the potential errors that can occur at each output of an 8-point FFT as well as the expected output error values (in all cases they are 0 except from Y0 due to orthogonality) and its complex variance. Since the complex variance is the sum of the variances of the real and imaginary parts and the expected values of the error at each output is 0 (except Y0), the complex variance is actually the sum of the squares of the sine and cosine of the same number (the power of the corresponding twiddle factor) which results in 1. Thus, the complex variance is equal to 1×ε

^{2}in all cases but Y0. This fact holds for any N-point FFT. If R is the number of samples substituted by the undersampling procedure (e.g., R=1/16 means that N/16 of the FFT inputs have been substituted by others), the total power of the UE error (P

_{UE}) can be estimated as a function of N, R, s, E[ε]:

_{f}(e.g., P

_{QE}, P

_{RE}≤ 10% of the P

_{UE}) of the UE error estimated in the way described above. More specifically, using ${P}_{RE}=8{\sigma}_{n}^{2}$ from Equation (11) and defining $c=1+{u}^{2}+{a}^{2}$, P

_{RE}can be expressed as:

^{nd}degree equation as follows:

_{f}, R, the word length b can be expressed as:

_{i}values, the Octave fsolve function for non-linear equations is used with a small number of instances of Equation (23) i.e., with a small number of b, p

_{f}, s, R, N, combinations. The experimental results show that Equation (23) can be used then to accurately estimate the required word length b for other p

_{f}values given a specific OFDM configuration (s, R, N, ε). The physical meaning of the estimated c

_{i}parameters will be explained in Section 5.

## 4. The Employed FFT Architecture

^{2}) operations that are reduced to O(N∙logN) if the original FFT architecture is employed [18]. The number of points used by the FFT can be expressed as a product of numbers that are powers of 2. Thus, a 1024-point FFT can be implemented by 10 Radix-2 stages, or 5 Radix-4 stages. If the number of points of the FFT is not a power of 2, then Radix-3 or Radix-5 butterflies can also be employed. For example, a 100-point FFT can be implemented with one stage of Radix-4 and two stages of Radix-5 butterflies [26]. The round-off errors depend on the architecture of the FFT (serial/parallel, Decimation in Time or Frequency, etc.) and the number of stages. An FFT can be implemented either in software if slower operation is acceptable or in hardware for faster response. Modern telecommunication systems require high speed hardware FFTs. Hardware FFTs can either consist of a large number of hardware resources working in parallel or reusable components for more compact, low power implementations with a slightly higher latency overhead.

_{2}N stages (one of them appears in Figure 5). The inputs of stage l are stored in the double buffer l (its size is 2×N×b bits). The word length of a butterfly output can be larger by one bit compared to its inputs for optimal resource utilization. However, we use a constant size of b-bits for the inputs/outputs and twiddles of all stages in order to get similar results with the case where a single reusable pipeline stage was used iteratively. One buffer l of the pair is used to store the real and the other for the imaginary part of the FFT inputs/outputs. Buffer l is accessed for write through the buses w1(l) and w2(l), and for read through the buses r1(l) and r2(l). Each one of these buses consists of a log

_{2}N bits, address bus (ra(l) or wa(l)) and a pair of b-bits data buses (Re{rd(l)} and Im{rd(l)}, or Re{rd(l)} and Im{rd(l)}). Each data bus carries real numbers in fixed point format with a size of b bits. The inputs of each Radix-2 butterfly are the rd1(l) and rd2(l) while its outputs are wd1(l) and wd2(l). The real and imaginary parts of the twiddle factors w are retrieved from the twiddle Read Only Memory (ROM). The size of the twiddle ROM of stage l is 2×N/2

^{l+1}.

_{2}(N)-1 resolution. In each stage l the pair of addresses used for the retrieval of the butterfly inputs/outputs (Addr0 for I

_{0}and O

_{0}, Addr1 for I

_{1}, O

_{1}) and AddrT for the corresponding twiddle factor are the following:

## 5. Simulation Results and Discussion

_{i}parameters. Then, the rest of the L=116 configurations were tested and the RMSE between the real value of b and the estimated b

_{est}for a specific p

_{f}value is extracted as shown in Equation (31). In this way, the minimum number of non-linear equations that have to be solved in order to determine the c

_{i}parameters precisely is found.

_{i}parameters. The average RMSE achieved in the word length estimation of all the 116 configurations is also listed in the 1

^{st}row of these tables along with the estimated c

_{i}values for each case. As can be seen from Table 3, determining the c

_{i}values from 12 instances of Equation (23) leads to the lowest RMSE (0.736 for 16QAM and 1.09 for QPSK modulation). A relatively low RMSE is also estimated if 16 equations are used as shown in Table 2. The c

_{i}parameters estimated in Table 2 and Table 3 are rounded to c

_{1}= −5, c

_{2}= −2, c

_{3}= 2, c

_{4}= 1, c

_{5}= 2 in order to explain the physical meaning of these values and how they lead to an accurate word length estimation.

_{i}values, Equation (23) can be written as:

_{1}will be ignored in an attempt to define the overall error model that matches the experimental results more accurately. In this perspective, the rest of the terms in the right side of Equation (23) are interpreted as follows: $\frac{1}{2}{\mathrm{log}}_{2}3RE[\epsilon ]$ takes its minimum value (−0.185) for the experiments conducted in this paper when R=1/16 and E[ε]=2.276 with QPSK modulation and its maximum value (0.2) with R=1/4 and E[ε]=3.37 when 16QAM modulation is employed. The term ${s}^{-2}{p}_{f}^{2}$ is close to a constant since p

_{f}is proportional to the sparse level s: if only non-sparse FFT inputs were present, there would be errors in all the FFT outputs and ${p}_{ns}={P}_{QE}/{P}_{UE}$. The higher value measured for p

_{ns}is 0.3. If the input is sparse, the ratio p

_{f}of the quantization error to the undersampling error is proportional to the sparseness level s: ${p}_{f}=\propto {p}_{ns}s$. When the input is too sparse, the UE and QE errors are both low. When the input is less sparse (s value is higher), UE raises but the raise of QE is even higher. This is owed to the fact that although UE gets worse, there may be still FFT outputs unaffected by the undersampling process if some samples are replaced by the others with identical value. However, if s is higher, more operations with numbers that are not zero will be performed and the QE will increase respectively since all the results of these non-trivial operations will have QE error. In this sense, ${s}^{2}$ counterbalances ${p}_{f}^{2}$ and the maximum value for the 3rd term of Equation (23) will be $\frac{1}{2}{\mathrm{log}}_{2}{s}^{-2}{p}^{2}=\frac{1}{2}{\mathrm{log}}_{2}{p}_{ns}=\frac{1}{2}{\mathrm{log}}_{2}0.3=-0.87$. If p

_{ns}is lower, a higher positive offset in Equation (23) occurs.

_{1}. The specific c

_{i}parameters have been approximated for these two FFT sizes. Should different FFT sizes be covered, the set of nonlinear equations that have to be used for the approximation of c

_{i}parameters must also include configurations with these FFT sizes. If we try to use the approximated c

_{i}parameters of Table 3 for the case of a 64-point FFT size, ${c}_{1}+{\mathrm{log}}_{2}N$ would be 0. The term $-\frac{1}{2}{\mathrm{log}}_{2}\left(3\xb7R\xb7E[\epsilon ]\right)$ results in small signed offsets between −0.185 and 0.2 as explained above and thus, the word length would be actually determined by the factor $-\frac{1}{2}{\mathrm{log}}_{2}{s}^{-2}{p}_{f}^{2}=-\frac{1}{2}{\mathrm{log}}_{2}{p}_{ns}$. In order to get a realistic estimation of at least 5 bits as a word length, p

_{ns}should be 10

^{-3}, or, in other words, UE error should be 1000 times larger than QE error. Such a relation between UE and QE errors is not always guaranteed.

_{i}parameters listed in Table 3 are used, are compared in Figure 6 and Figure 7, respectively. In these figures, the required minimum ADC resolution is also included. This ADC resolution b

_{ADC}has been estimated in Equation (33) that has been derived from Equation (8), the definition $\mathit{\Delta}={V}_{ref}/{2}^{{b}_{ADC}}$ and the specification that P

_{QE}should be equal to P

_{RE}. V

_{ref}was selected equal to 1V but approximately the same results would have been achieved if a different voltage reference had been selected, such as 3V. The ADC resolution b

_{ADC}should match the FFT word length b thus, b

_{ADC}should be selected equal to b since ${b}_{ADC}<b$ in all cases as shown in Figure 6 and Figure 7.

_{s}) in order to achieve a desired capacity, given a specific power budget P

_{o}. Based on the analysis presented in [12], the capacity C

_{o}can be expressed as:

_{o}R

_{s}can be expressed as a weighted combination of the various noise sources in an optical channel [12]: the beat noise ${\sigma}_{beat}^{2}=4{S}_{b}^{2}{N}_{o}{P}_{LO}{B}_{e}$, the shot noise ${\sigma}_{shot}^{2}=2e{S}_{b}^{}{P}_{LO}{B}_{e}$ and the thermal/electronic noise ${\sigma}_{elec}^{2}$. Concerning the constants (we assume that their values are known) used in these noise variance expressions, S

_{b}is the photodetector responsivity, N

_{o}is the noise spectral density, P

_{LO}the optical power, B

_{e}the power equivalent bandwidth of the entire receiver and e the elementary charge. The most important optical noise is Amplified Spontaneous Emission (ASE) which actually describes the attenuation of the optical signal by a factor ${a}_{ASE}=0.2dB/Km$. The distortion posed by the required N

_{A}repeater/amplifiers placed at distance ${L}_{a}$ can be described by the noise spectral density ${N}_{ASE}^{EDRA}={N}_{A}\left({e}^{{a}_{ASE}{L}_{a}}-1\right)h{v}_{s}{n}_{sp}$ or ${N}_{ASE}^{IDRA}={a}_{ASE}{L}_{a}h{v}_{s}{K}_{T}$ if Erbium-doped fiber amplifiers (EDFAs) or Ideal Distributed Raman Amplification (IDRA) is used, respectively. The parameters used in these noise spectral densities are also assumed to have known values: h is the Plank constant, v

_{s}is the optical frequency, K

_{T}is the photon occupancy factor and ${n}_{sp}<1$ the spontaneous emission factor. The model can be trained by a number of non-linear equations that combine the channel error sources with various capacity and power requirements for specific predefined modulations schemes. The target of this training would be to estimate the weights of the channel error sources. After updating the error model with these weights, it can be used to select an appropriate modulation scheme for different capacity and power specifications or channel conditions.

## 6. Conclusions

## 7. Patents

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 3.**Error propagation from top (

**a**) or bottom (

**b**) of the butterfly to the next stage of a DIT FFT.

**Figure 6.**Estimated, expected word length and minimum ADC resolution for all the 16QAM OFDM configurations tested.

**Figure 7.**Estimated, expected word length and minimum ADC resolution for all the QPSK OFDM configurations tested.

Path: | 000 (×ε) | 001 (×ε) | 010 (×ε) | 011 (×ε) | 100 (×ε) | 101 (×ε) | 110 (×ε) | 111 (×ε) | $\mathbf{Expected}\text{}\mathbf{Error}\text{}\left(\text{\xd7}\mathbf{E}\right[\mathsf{\epsilon}\left]\right),\text{}\mathbf{Variance}\text{}{\mathit{\sigma}}_{\mathit{U}\mathit{S}}^{2}\left(\mathit{k}\right)\text{}(\times \mathbf{E}[\mathsf{\epsilon}{]}^{2})$ |
---|---|---|---|---|---|---|---|---|---|

Y_{0} | 1 | ${w}_{8}^{0}$ | ${w}_{8}^{0}$ | ${w}_{8}^{0}{w}_{8}^{0}$ | ${w}_{8}^{0}$ | ${w}_{8}^{0}{w}_{8}^{0}$ | ${w}_{8}^{0}{w}_{8}^{0}$ | ${w}_{8}^{0}{w}_{8}^{0}{w}_{8}^{0}$ | 1, 0 |

Y_{1} | 1 | ${w}_{8}^{1}$ | ${w}_{8}^{2}$ | ${w}_{8}^{2}{w}_{8}^{3}$ | $-{w}_{8}^{0}$ | $-{w}_{8}^{0}{w}_{8}^{1}$ | $-{w}_{8}^{0}{w}_{8}^{2}$ | $-{w}_{8}^{0}{w}_{8}^{2}{w}_{8}^{3}$ | 0, 1 |

Y_{2} | 1 | ${w}_{8}^{2}$ | -${w}_{8}^{0}$ | -${w}_{8}^{0}{w}_{8}^{2}$ | ${w}_{8}^{0}$ | ${w}_{8}^{0}{w}_{8}^{2}$ | $-{w}_{8}^{0}{w}_{8}^{0}$ | $-{w}_{8}^{0}{w}_{8}^{0}{w}_{8}^{2}$ | 0, 1 |

Y_{3} | 1 | ${w}_{8}^{3}$ | $-{w}_{8}^{2}$ | $-{w}_{8}^{2}{w}_{8}^{3}$ | $-{w}_{8}^{0}$ | $-{w}_{8}^{0}{w}_{8}^{3}$ | ${w}_{8}^{0}{w}_{8}^{2}$ | ${w}_{8}^{0}{w}_{8}^{2}{w}_{8}^{3}$ | 0, 1 |

Y_{4} | 1 | $-{w}_{8}^{0}$ | ${w}_{8}^{0}$ | $-{w}_{8}^{0}{w}_{8}^{0}$ | ${w}_{8}^{0}$ | $-{w}_{8}^{0}{w}_{8}^{0}$ | ${w}_{8}^{0}{w}_{8}^{0}$ | $-{w}_{8}^{0}{w}_{8}^{0}{w}_{8}^{0}$ | 0, 1 |

Y_{5} | 1 | $-{w}_{8}^{1}$ | ${w}_{8}^{2}$ | $-{w}_{8}^{2}{w}_{8}^{1}$ | $-{w}_{8}^{0}$ | ${w}_{8}^{0}{w}_{8}^{1}$ | $-{w}_{8}^{0}{w}_{8}^{2}$ | ${w}_{8}^{0}{w}_{8}^{2}{w}_{8}^{1}$ | 0, 1 |

Y_{6} | 1 | $-{w}_{8}^{2}$ | $-{w}_{8}^{0}$ | ${w}_{8}^{0}{w}_{8}^{2}$ | ${w}_{8}^{0}$ | $-{w}_{8}^{0}{w}_{8}^{2}$ | $-{w}_{8}^{0}{w}_{8}^{0}$ | ${w}_{8}^{0}{w}_{8}^{0}{w}_{8}^{2}$ | 0, 1 |

Y_{7} | 1 | $-{w}_{8}^{3}$ | $-{w}_{8}^{2}$ | ${w}_{8}^{2}{w}_{8}^{3}$ | $-{w}_{8}^{0}$ | ${w}_{8}^{0}{w}_{8}^{3}$ | ${w}_{8}^{0}{w}_{8}^{2}$ | $-{w}_{8}^{0}{w}_{8}^{2}{w}_{8}^{3}$ | 0, 1 |

RMSE: 0.816 (16QAM), 1.132 (QPSK), c_{1} = −5.6315, c_{2} = −2.0292, c_{3} = 2.0669, c_{4} = 1.1994, c_{5} = −2.3163 | ||||||
---|---|---|---|---|---|---|

Equation | Word Length (b) | p_{f}=P_{QE}/P_{UE} | QAM Modula-tion | FFT Size N | Number of Substituted Samples R | Sparseness s |

1 | 5 | 0.0046 | 16QAM | 1024 | 1/4 | 0.5% |

2 | 8 | 0.000344 | 16QAM | 1024 | 1/4 | 0.5% |

3 | 5 | 0.046 | 16QAM | 256 | 1/4 | 10% |

4 | 8 | 0.002624 | 16QAM | 256 | 1/4 | 10% |

5 | 5 | 0.12 | 16QAM | 1024 | 1/16 | 10% |

6 | 10 | 0.00773 | 16QAM | 1024 | 1/16 | 10% |

7 | 5 | 0.002625 | 16QAM | 256 | 1/16 | 0.5% |

8 | 8 | 0.00026 | 16QAM | 256 | 1/16 | 0.5% |

9 | 5 | 0.00377 | QPSK | 1024 | 1/4 | 0.5% |

10 | 10 | 0.001 | QPSK | 1024 | 1/4 | 0.5% |

11 | 5 | 0.0245 | QPSK | 256 | 1/4 | 10% |

12 | 10 | 0.000519 | QPSK | 256 | 1/4 | 10% |

13 | 5 | 0.15 | QPSK | 1024 | 1/16 | 10% |

14 | 10 | 0.036 | QPSK | 1024 | 1/16 | 10% |

15 | 5 | 0.0016 | QPSK | 256 | 1/16 | 0.5% |

16 | 10 | 0.000273 | QPSK | 256 | 1/16 | 0.5% |

RMSE: 0.736 (16QAM), 1.09 (QPSK), c_{1} = −4.3811, c_{2} = −1.8297, c_{3} = 1.9560, c_{4} = 1.2176, c_{5} = −2.0358 | ||||||
---|---|---|---|---|---|---|

Equation | Word Length (b) | p_{f}=P_{QE}/P_{UE} | QAM Modula-tion | FFT Size N | Number of Substituted Samples R | Sparseness s |

1 | 8 | 0.000344 | 16QAM | 1024 | 1/4 | 0.5% |

2 | 5 | 0.046 | 16QAM | 256 | 1/4 | 10% |

3 | 8 | 0.002624 | 16QAM | 256 | 1/4 | 10% |

4 | 5 | 0.12 | 16QAM | 1024 | 1/16 | 10% |

5 | 5 | 0.002625 | 16QAM | 256 | 1/16 | 0.5% |

6 | 8 | 0.00026 | 16QAM | 256 | 1/16 | 0.5% |

7 | 5 | 0.00377 | QPSK | 1024 | 1/4 | 0.5% |

8 | 10 | 0.001 | QPSK | 1024 | 1/4 | 0.5% |

9 | 10 | 0.000519 | QPSK | 256 | 1/4 | 10% |

10 | 5 | 0.15 | QPSK | 1024 | 1/16 | 10% |

11 | 10 | 0.036 | QPSK | 1024 | 1/16 | 10% |

12 | 10 | 0.000273 | QPSK | 256 | 1/16 | 0.5% |

RMSE: 1.118 (16QAM), 1.62 (QPSK), c_{1} = 4.00823, c_{2} = −1.29494, c_{3} = 1.55213, c_{4} = 1.04125,c_{5} = −0.16769 | ||||||
---|---|---|---|---|---|---|

Equation | Word Length (b) | p_{f}=P_{QE}/P_{UE} | QAM Modulation | FFT Size N | Number of Substituted Samples R | Sparseness s |

1 | 8 | 0.000344 | 16QAM | 1024 | 1/4 | 0.5% |

2 | 5 | 0.046 | 16QAM | 256 | 1/4 | 10% |

3 | 5 | 0.12 | 16QAM | 1024 | 1/16 | 10% |

4 | 8 | 0.00026 | 16QAM | 256 | 1/16 | 0.5% |

5 | 5 | 0.00377 | QPSK | 1024 | 1/4 | 0.5% |

6 | 10 | 0.000519 | QPSK | 256 | 1/4 | 10% |

7 | 5 | 0.15 | QPSK | 1024 | 1/16 | 10% |

8 | 10 | 0.000273 | QPSK | 256 | 1/16 | 0.5% |

RMSE: 3.694 (16QAM), 2.622 (QPSK), c_{1} = −33.2488, c_{2}=−3.6799, c_{3} = 4.8570, c_{4} = 4.9535, c_{5} = −5.9386 | ||||||
---|---|---|---|---|---|---|

Equation | Word Length (b) | p_{f}=P_{QE}/P_{UE} | QAM Modulation | FFT Size N | Number of Substituted Samples R | Sparseness s |

1 | 8 | 0.002624 | 16QAM | 256 | 1/4 | 10% |

2 | 5 | 0.002625 | 16QAM | 256 | 1/16 | 0.5% |

3 | 5 | 0.00377 | QPSK | 1024 | 1/4 | 0.5% |

4 | 5 | 0.15 | QPSK | 1024 | 1/16 | 10% |

5 | 10 | 0.036 | QPSK | 1024 | 1/16 | 10% |

© 2020 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Petrellis, N.
Selecting FFT Word Length for an OFDM Receiver That Supports Undersampling. *Symmetry* **2020**, *12*, 543.
https://doi.org/10.3390/sym12040543

**AMA Style**

Petrellis N.
Selecting FFT Word Length for an OFDM Receiver That Supports Undersampling. *Symmetry*. 2020; 12(4):543.
https://doi.org/10.3390/sym12040543

**Chicago/Turabian Style**

Petrellis, Nikos.
2020. "Selecting FFT Word Length for an OFDM Receiver That Supports Undersampling" *Symmetry* 12, no. 4: 543.
https://doi.org/10.3390/sym12040543