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ORTHOGONAL FREQUENCY DIVISION MULTIPLEXIONG (OFDM) 1  ORTHOGONAL FREQUENCY DIVISION MULTIPLEXIONG (OFDM) 1. Introduction Orthogonal frequency-division multiplexing (OFDM) is a multi-carrier signaling technique that is best categorized as a combination of tw...

ORTHOGONAL FREQUENCY DIVISION MULTIPLEXIONG (OFDM) 1  ORTHOGONAL FREQUENCY DIVISION MULTIPLEXIONG (OFDM) 1. Introduction Orthogonal frequency-division multiplexing (OFDM) is a multi-carrier signaling technique that is best categorized as a combination of two operations: modulation and multiplexing (or multiple access). It is a special case of FDM but: o The carrier frequencies of the multiplexed signals are orthogonal. o It saves bandwidth since it allows frequency overlap between neighboring signals. o Demultiplexing uses no filtering but orthogonality properties. OFDM is such a valuable technique for mitigating the degrading effects of fading channels in wireless communications. The problem of fading channels becomes serious for high data rates. The basic idea of OFDM is to partition a high-data-rate signal into smaller low-data- rate signals so that the data can be sent over many low-rate subchannels. Discrete Fourier transforms (DFTs) are used for efficient baseband modulation and demodulation. Thus, they eliminate the need for banks of oscillators. 2. FDM versus OFDM Consider a single carrier signal with symbol rate 𝑅𝑠. It is required to use multicarrier system to transmit the same data Assume the number of carriers is 𝑁𝐶. In conventional multi-channel FDM system shown in Figure 1a: o Such systems have nonoverlapping adjacent channels. o The symbol rate per subchannel is 𝑅𝑆𝐶 = 𝑅𝑆 ⁄𝑁𝐶 = 1⁄𝑁𝐶 𝑇𝑆 o The subchannel null-to-null bandwidth is 𝐵𝑆𝐶 = 2𝑅𝑆𝐶 = 2 𝑅𝑆 ⁄𝑁𝐶 = 2⁄𝑁𝐶 𝑇𝑆 o There are guard bands between subchannels so that they can be separated using filters. o The frequency separation between subchannels is ∆𝑓 > 𝐵𝑆𝐶 = 2 𝑅𝑆 ⁄𝑁𝐶 = 2⁄𝑁𝐶 𝑇𝑆 (1a) o The total required bandwidth is 𝑊 = 𝑁𝐶 ∆𝑓 > 𝑁𝐶 𝐵𝑆𝐶 = 2𝑅𝑆. (1b) In an OFDM multi-channel system shown in Figure 1b: o As in FDM, the symbol rate per subchannel is 𝑅𝑆𝐶 = 𝑅𝑆 ⁄𝑁𝐶 = 1⁄𝑁𝐶 𝑇𝑆 o Unlike FDM, there is no guard bands between subchannels since they are not separated using filters. o Moreover, in OFDM, there is a 50% overlap of adjacent channels. o The frequency separation between subchannels assuming noncoherent detection is ∆𝑓 = 𝑅𝑆𝐶 = 𝑅𝑆 ⁄𝑁𝐶 = 1⁄𝑁𝐶 𝑇𝑆 (2a) o The total required bandwidth is 𝑊 = 𝑁𝐶 ∆𝑓 = 𝑅𝑆 = 1⁄𝑇𝑆. (2b) Thus, OFDM has better bandwidth efficiency than a conventional system. 2  However, OFDM’s greatest advantage is not bandwidth saving but its contribution to communications over multipath fading channels is that it can greatly simplify the equalization task into one of simple scaling. Figure 1 Conventional multi-channel system versus OFDM multi-channel system. Such overlap is possible because the OFDM channels are orthogonal to each other. In OFDM, it is important to keep orthogonality among symbols, where the correlation between different symbols equal to zero, as described by 𝑇 ∫0 𝑠𝑖 (𝑡)𝑠𝑗 (𝑡) 𝑑𝑡 = 0 𝑖≠𝑗 (3) 3. OFDM in Fading Channels Coherence bandwidth 𝑓0 is a statistical measure of the range of frequencies that are treated by the channel in a like manner. It is the range of a signal’s spectral components traversing the channel, received with approximately equal gain and linear phase. Coherence time 𝑇𝑜 is a statistical measure of the range of time during which the signal is treated by the channel in a like manner. In multipath channel, there is a maximum delay 𝑇𝑚 between the longest path and the shortest path of arrival. Also motion causes frequency change (Doppler frequency shift) during the time of motion. It should be noted that the coherence bandwidth 𝑓𝑜 and maximum multipath spread 𝑇𝑚 are reciprocally related (within a multiplicative constant), as are the coherence time 𝑇𝑜 and the Doppler spread 𝑓𝑑 , i.e. 𝑓𝑜 ≈ 1⁄𝑇𝑚 𝑇𝑜 ≈ 1⁄𝑓𝑑 (4) 1 For frequency selective fading 𝑓𝑜 < 𝑊 or 𝑓𝑜 < (5a) 𝑇𝑠 1 For flat fading channel, 𝑓𝑜 > 𝑊 or 𝑓𝑜 > (5b) 𝑇𝑠 1 Fast Fading 𝑇𝑜 < 𝑇𝑠 or < 𝑓𝑑 (5c) 𝑇𝑠 1 For slow fading channel, 𝑇𝑜 > 𝑇𝑠 or > 𝑓𝑑 (5d) 𝑇𝑠 Worst-case fading profiles are typically characterized as frequency-selective fading and/or fast fading since they cause signal distortion and needs equalizers for proper signal recovery. 3  However flat fading and slow fading may cause attenuation and minimal distortion., that can be easily managed using traditional equalization techniques and diversity techniques. Therefore, when using OFDM (with the goal of avoiding worst-case fading), the channel wish list needs to be modified such that flat fading and slow fading channel are obtained as follows, 1 1 𝑓𝑜 > > (6a) 𝑇𝑠 𝑇0 𝑓𝑜 > 𝑊 > 𝑓𝑑 (6b) In other words, Coherence bandwidth > symbol rate > fading (Doppler) rate 𝑊 1 𝑊 𝑓𝑜 > > or 𝑓𝑜 > > 𝑓𝑑 (7) 𝑁𝐶 𝑇0 𝑁𝐶 where 𝑁𝐶 represents the number of subcarriers. In OFDM, our goal is to subdivide a frequency-selective channel into much smaller subchannels that do not suffer from any frequency-selective fading. The number of subcarriers 𝑁𝐶 is chosen such that frequency selective fading and fast fading are avoided. It should be noted that both coherence bandwidth and coherence time can be obtained from typical channel impulse response ℎ𝑐 (𝑡) for a channel, where 𝑇𝑚 is the maximum time spreading due to multipath. Figure 2 shows how OFDM technique mitigates frequency-selective-fading channel. Imagine a channel having a relatively narrow coherence bandwidth 𝑓0 , as shown in Figure 2a. It is required to transmit a signal of wide-bandwidth W (high-data rate), as shown in Figure 2b, over that channel, where 𝑊 > 𝑓0. In this case, the transmitted data will suffer from frequency selective fading, which results in signal distortion. Figure 2c is a sketch illustrating that OFDM’s mitigation stems from dividing the high-rate signal into a number 𝑁𝑐 of low-rate orthogonal subchannels. Each subchannel has a bandwidth 𝐵𝑆𝐶 = 𝑊 ⁄𝑁𝐶 < 𝑓0. Therefore, OFDM transforms one large frequency-selective-fading channel into many flat-fading subchannels. The delay spread is generally a statistical number and it can be best described by the rms delay spread 𝜎𝜏. The coherence bandwidth 𝑓𝑜 and the channel’s rms delay spread 𝜎𝜏 , assuming the channel response is within 0.5 of its maximum value, are related as: 1 𝑓𝑜 ≈ (8a) 5𝜎𝜏 𝑇𝑚 ≈ 5𝜎𝜏 (8b) Assuming the channel response is within 0.9 of its maximum value, then 1 𝑓𝑜 ≈ (9a) 50𝜎𝜏 𝑇𝑚 ≈ 50𝜎𝜏 (9b) 4  Figure 2 Mitigation of a frequency-selective-fading channel by the OFDM technique. (a) Channel frequency response. (b) Frequency-selective problem. (c) Mitigation. Example 1 OFDM Subcarrier Design Assume a mobile wireless channel with an rms delay spread 𝜎𝜏 =20 μsec and a coherence time 𝑇0 =10 ms. If the transmission bandwidth W is 100 kHz, find the minimum and maximum number of OFDM subcarriers needed to ensure flat fading and slow fading results. Find the maximum allowed velocity assuming the carrier frequency is 300 MHZ. Solution The coherence bandwidth 𝑓0 is 𝑓0 ≈ 1/5𝜎𝜏 = 1/ (5 × 20 × 10−6 ) = 10 kHz The minimum and maximum number of required subcarriers: (𝑁𝑐 )𝑚𝑖𝑛 ≥ 𝑊/𝑓0 = 100 kHz/10 khz = 10 The maximum number of required subcarriers: (𝑁𝑐 )𝑚𝑎𝑥 ≤ 𝑇0 𝑊 = 10−2 × 105 = 1000 The number of subcarriers should fall in the range of 10 to 1000. Note that the upper limit of 1000 subcarriers is dictated by the channel coherence time 𝑇0 (10 ms in this example), which is a function of the user’s velocity. To get an idea of the velocity corresponding to 𝑇0 = 10 ms, For carrier frequency of 300 MHz, the wavelength is 𝑐 3 × 108 𝜆= = =1𝑚 𝑓 300 × 106 The velocity can be given by the relation 𝜆 1 𝑇0 = = = 10−2 𝑠 2𝑣 2𝑣 Thus, the velocity is 𝑣 = 50 𝑚/𝑠 = 180 𝑘𝑚/ℎ𝑟 5 Example 2 Modem Without OFDM and With OFDM Consider a wireless radio system with a data rate R = 1 Mbit/s that is channel encoded with a 1/2 code rate to provide coded bits with a rate 𝑅𝐶 = 2 × 106 𝑐𝑜𝑑𝑒𝑑 𝑏𝑝𝑠. Using 8-PSK modulation, the symol rate is 𝑅𝑆 = 2/3 M symbols/s as shown in Figure 3. Assume that the channel’s rms delay spread 𝜎𝜏 =3 μsec. (a) Without the benefit of OFDM, find the rms delay 𝜎𝜏 in units of symbol intervals. Also find the maximum delay 𝑇𝑚 (in the same symbol-interval units). (b) Repeat these computations with OFDM when the number of subcarriers 𝑁𝐶 = 64. Solution We are given the bit-time 𝑇𝑏 = 1 μsec and the channel rms-delay spread 𝜎𝜏 = 3 μsec. (a) Rate of coded bits 𝑅𝐶 = 2𝑅𝑏 due to 1/2 code rate Symbol rate 𝑅𝑆 = 𝑅𝐶 ⁄3 = 2𝑅𝑏 ⁄3 due to 8-PSK modulation symbol duration 𝑇𝑆 = 1/𝑅𝑆 = (3/2𝑅𝑏 ) = 3/2 μsec. rms delay 𝜎𝜏 =3 μsec = 2 𝑇𝑆. The maximum multipath spread 𝑇𝑚 ≈ 5𝜎𝜏 = 15 μsec = 10𝑇𝑆 There is a large overlap (ISI) between neighboring symbols Figure 3 Modem without OFDM and with OFDM (b) The goal of OFDM is to lengthen the time duration of the transmitted symbol so that the portion of the symbol being smeared due to channel-induced ISI is reduced or mitigated. In this example, the high-data-rate input is divided into multiple lower-data-rate subchannels (with a lengthened symbol interval 𝑇𝑂𝐹𝐷𝑀 ). For sampled systems, the transmitter algorithm that efficiently transforms a high-data rate, short-symbol input into a lower-data-rate, lengthened-symbol output is the inverse discrete Fourier transform (IDFT), denoted as 64-point IDFT. Note that without OFDM, 𝑇𝑆 = (3/2) μsec, and the rms smearing occupies 200 % of a symbol time. 6  But with OFDM, the lengthened 𝑇𝑂𝐹𝐷𝑀 = 𝑁𝑐 𝑇𝑂𝐹𝐷𝑀 = =64 × 𝑇𝑆 = 96 μsec. Hence the rms smearing (𝜎𝜏 = 3 μsec) only occupies ≈ 3% of the lengthened symbol time. The maximum multipath spread 𝑇𝑚 ≈ 5𝜎𝜏 = 15 μsec = 0.156𝑇𝑆 Therefore, the longer symbols in OFDM have the effect of significantly lowering the corruptive smearing effects of channel-induced ISI. 4. The Cyclic Prefix (CP) In OFDM, the lengthened symbols greatly reduce the overlap between symbols (ISI). Guard periods of duration equal to the maximum delay spread are introduced between symbols such that no ovelap between pulses takes place. The cyclic prefix (CP) was introduced to maintain the orthogonality of received signals that had been subjected to severe multipath conditions whenever the CP is longer than the channel impulse response. The use of CP allows circular convolution, which provides equalization for the dispersive channels in a simple way. Figure 4 shows some details in the formation of the elongated OFDM symbols. The channel has impulse response ℎ𝑐 (𝑡). In Figure 4a: o OFDM is not used. o Data symbols are represented as impulses o Channel delay spread is much larger than the symbol duration o The individual ℎ𝑐 (𝑡) effects are summed (super-positioned) and the received symbols are smeared resulting in channel-induced ISI. o This distortion is formed by convolving the data-symbol impulses with ℎ𝑐 (𝑡). In Figure 4b: o A high-rate signal is portioned into many low rate subchannels. o Each partition has a much lower bandwidth than the channel coherence bandwidth 𝑓𝑜. o Thus, each data symbol is lengthened by a large factor. o Convolving such lengthened symbols with the same ℎ𝑐 (𝑡) still yields distortion. o But because the symbol is now longer, the portion being corrupted by smearing is significantly reduced. o Thus, each partition suffers only negligible frequency-selective fading. Figure 4c shows the ideal adjacent symbols transmitted and Figure 4d shows their distorted versions received. It is easy to fix this distortion by paying the price of separating the symbols with guard intervals before transmission as shown in in Figure 4e. Figure 4f shows the received symbols separated by the guard intervals. These waveforms are still distorted, but there is no smearing. Figures 4g and 4h introduce the “very clever” cyclic prefix (CP), which utilizes the guard interval to do more than just remove any distortion effects of ISI. 7  The CP is formed in Figure 6g by replicating the back end of the waveform to be transmitted and introducing it at the front end, within the guard interval. The CP helps to convert linear convolution (with the channel impulse response) into circular convolution, thereby preserving signal orthogonality. It does this by ensuring that the receiver sees a constant-envelope signal having an integer number of cycles. Such action makes the task of equalization simple (spectral scaling). Once these modified symbols are received, as seen in Figure 4h, the CP is discarded by the receiver, leaving the same uncorrupted symbols sent in Figure 4e. To ensure that any channel-induced ISI is totally mitigated at the receiver after the CP is discarded, the length 𝑇𝐶𝑃 of the CP should be longer than the maximum delay spread 𝑇𝑚. 𝑇𝐶𝑃 > 𝑇𝑚 Because increasing the CP length reduces the channel’s capacity, a rule of thumb for choosing the CP length is to use a length two to four times the channel’s rms delay spread. 𝑇𝐶𝑃 ≈ (2 − 4)𝜎𝜏 The smaller multiplier yields a larger link capacity but more frequently occurring channel-induced ISI. The larger multiplier yields a smaller capacity but less frequently occurring channel- induced ISI. Figure 4 Examining details in the formation of lengthened OFDM symbols. Note that 𝑇𝑂𝐹𝐷𝑀 = 𝑇𝑆 + 𝑇𝐶𝑃. 8 5. OFDM System Block Diagram Figure 5 is a simplified transmitter/receiver OFDM block diagram. The figure’s top half shows the transmitter, where the OFDM signal is formed. The bottom half shows the receiver, where the OFDM formation is essentially reversed. Serial-to-parallel (S/P): o It converts a serial stream of high-rate data symbols 𝑅𝑏 into 𝑁𝐶 parallel streams of lower data rate, where 𝑁𝐶 is the number of parallel channels or subcarriers. o Thus, the data rate of each parallel stream is 𝑅𝑏 ⁄𝑁𝐶 o Each parallel channel is assigned a different subcarrier. Constellation mapping: o It is called baseband modulation or data modulation. o Each successive 𝑚 bits in a channel is mapped into a phasor according to the phasor (or constellation) diagram of the modulation types used. o For example, in BPSK, the phasors are 1 and -1 (which are real) for bits 1 and 0, respectively. o For QPSK, the phasors are (1+j), (-1+j), (-1-j), and (1-j) for the dibits 11, 01, 00, and 10, repectively. o Thus, we have 𝑁𝐶 parallel phasors, {𝑑𝑛 }, where each phasor is assigned a different subcarrier. o These phasors represent a total of 𝑚𝑁𝐶 bits. o For example, using 64-QAM, i.e. 𝑚 = 6 bits and assuming 𝑁𝐶 = 50, then each phasor represents 6 bits and the set of parallel phasors represent 300 bits. IDFT (Inverse Discrete Fourier Transform): o Since the phasors are assigned different subcarriers, the set of parallel phasors represent the frequency domain. o The phasors are presented in parallel to the input of an N-point (IDFT), where 𝑁 > 𝑁𝐶 and 𝑁 is a power of two. o The output of the IDFT is the time domain of the composite symbol that consists of superimposed 𝑁𝐶 symbols, each has its own frequency, amplitude, and phase. o The subcarriers should be orthogonal and, therefore, they are spaced apart from their neighbors by 𝛥𝑓 given by 𝛥𝑓 = 𝑅𝑠 = 1⁄𝑇𝑠 (10a) o 𝑇𝑠 is the time duration of the data portion of the OFDM symbol. o The output of IDFT at each bin is a sample of the composite signal and it has contributions from all the sinusoids of the subcarriers. o We can think of the IDFT as a signal generator, with constellation points (assigned to frequency bins) at the input and time waveforms at the output. o The complex constellation point of each data symbol represents the magnitude and phase of the (sin x)/x spectrum of its assigned subcarrier. o At its output, the IDFT frequency-to-time mapping yields the coefficients of a sinusoidal basis set that describe the magnitude, phase, and frequency of each sinusoid to be built. o Since these output sinusoids have rectangular envelopes, we call them gated sinusoids. 9 o In order to form an OFDM symbol, the cyclic prefix of duration 𝑇𝐶𝑃 is appended to the waveform obtained from the IDFT output. o The time duration of the OFDM symbol is 𝑇𝑂𝐹𝐷𝑀 = 𝑇𝑠 + 𝑇𝐶𝑃 (10b) o 𝑇𝐶𝑃 should be larger than the maximum delay spread, 𝑇𝑚 , of the channel. o It is obvious that the CP is an overhead that will be discarded in the receiver, leaving the data portion 𝑇𝑠 of the symbol. o the transmission efficiency 𝑇 𝑇𝑆 1 𝜂= 𝑆 = = ⁄ (10c) 𝑇𝑂𝐹𝐷𝑀 𝑇𝑆 + 𝑇𝐶𝑃 1+𝑇𝐶𝑃 𝑇𝑆 o In order to increase the transmission efficiency, 𝑇𝑠 should be several times larger than the maximum delay spread of the channel. 1 1 1 1 o Generally, 𝑇𝐶𝑃 may be , , , , … of the duration of the data portion 𝑇𝑆. 4 8 16 32 o Figure 5 shows where the CP is added (at the transmitter’s IDFT output) and where it is removed (at the receiver’s DFT input). o The output of the IDFT has N samples within the data symbol duration 𝑇𝑠. o The number of samples within the CP duration is 𝑁𝐶𝑃. o The total number of samples within OFDM symbol is 𝑁𝐿 = 𝑁 + 𝑁𝐶𝑃 (11a) o Thus, the sampling rate is 𝑁 𝑁 𝑓𝑆 = = 𝐿. (11b) 𝑇𝑠 𝑇𝑂𝐹𝐷𝑀 o The sampling interval is 1 𝑇 𝑇𝐿 = = 𝑂𝐹𝐷𝑀 (12a) 𝑓𝑠 𝑁𝐿 o The OFDM symbol duration can be expressed as 𝑇𝑂𝐹𝐷𝑀 = 𝑁𝐿 𝑇𝐿 (12b) o Figure 6 represents Data constellation points (16-QAM) distributed over time- frequency indexes for three OFDM symbols. 10  Figure 5 OFDM transmitter/receiver block diagram. DAC (digital to analog Converter) o The real and imaginary parts of the time domain composite symbol are separated. o They are converted to analog signals. o Thus, they are interpolated to a continuous waveform. o This process is necessary for bandpass modulation. Figure 6 Data constellation points (16-QAM) distributed over time-frequency indexes. 11  LPF (low pass Filter) o It is used to limit the signal bandwidth of the real and imaginary parts of the baseband composite symbol. o This process is necessary to avoid interference. Bandpass Modulation o Quadrature modulation is used for both the real and imaginary parts of the composite baseband symbol. o The real RF signal is 𝑠(𝑡) = 𝑥(𝑡)𝑐𝑜𝑠 𝜔𝑐 𝑡 + 𝑦(𝑡) 𝑠𝑖𝑛 𝜔𝑐 𝑡 where 𝑥(𝑡) and 𝑦(𝑡) are the real and imaginary parts. Power Amplification o The signal 𝑠(𝑡) is power amplified o The power level should consider the area coverage and required error performance. Signal Processing in the OFDM Receiver o In the OFDM receiver, the received signal is first quadrature down-converted using coherent demodulation and carrier 𝑓𝐶 recovery. o The baseband signals are time sampled to form the real and imaginary parts of the composite symbol. o The samples of the cyclic prefix is removed to leave N-length block of such samples o Then, the reverse process (DFT) converts each N-length block of such samples to a sequence of 𝑁𝐶 nonzero spectral coefficients. o The spectral coefficients should be the same as the phasers generated by the mappers at the transmitter. o Knowing the phasors ( constellation points) the bit combination representing this phasor can be obtained. o For emphasis, we repeat that the actual OFDM signal processing never displays continuous sinusoids or sinc functions but only samples. The Cyclic Prefix (CP) and Tone Spacing As seen in Figure 7a, for a sinusoidal data waveform length T (with rectangular envelope), its spectral (sin x)/x or sinc width is defined as 1/T. For orthogonal spacing of such subcarrier tones, this 1/T width is the same as the distance between adjacent spectral peaks and accounts for the 50% spectral overlap, as seen in the figure. In Figure 7b, the data length is extended to 𝑇𝑋 = 𝑇 + 𝑇𝐸𝑋𝑇 (as in the case of a CP). Therefore, the sinc width is reduced to 1/𝑇𝑋. Note that the subcarrier spacing is not reduced to be 1/𝑇𝑋 , but preserved 1/𝑇 because we haven’t changed the cycles per interval (frequency). Thus, Orthogonality in “space” is violated but we are concerned about orthogonality of the waveforms at the receiver, after the extension is discarded. 12 Figure 7(a) Data length defines sinc width: spectral spacing matches width. (b) Extended data length reduces sinc width, but spectral spacing is preserved. Example 3: OFDM Waveform Synthesis Figure 8 shows how the gated sinusoids are formed and combined to yield the output OFDM waveform. Consideran IDFT with 𝑁 = 8 and 𝑁𝐶 = 4. Thus, the number of null bins is 𝑁𝑛𝑢𝑙𝑙 = 𝑁 − 𝑁𝐶 = 4 The output from each IDFT bin is a time waveform dictated by a data symbol that has modulated a subcarrier at center frequency n, expressed as 2𝜋 𝑗( 𝑛)𝑘 𝑑𝑛 𝑒 𝑁 where k and n are the time and subcarrier indexes, respectively. The output OFDM waveform samples {x(k)} are formed from the superposition of all the N subcarriers at each moment of time k, as follows: 2𝜋 𝑗( 𝑁 𝑛 )𝑘 𝑥𝑘 = ∑𝑁−1 𝑛=0 𝑑𝑛 𝑒 𝑛, 𝑘 = 0, 1, 2,... , 𝑁 – 1 2𝜋 In our example, 𝑥𝑘 = ∑𝑁−1 𝑛=0 𝑑𝑛 𝑒 𝑗( 𝑁 𝑛 )𝑘 𝑛, 𝑘 = 0, 1, 2,... ,7 and 𝑑𝑛 = 0 for 𝑛 = 4, 5, 6,7. In the bottom-left portion of the figure, we see four subcarriers, made up of a DC term, one cycle of a fundamental sinusoid, and second and third harmonics. Each subcarrier is multiplied by its associated data phasor. The data-phasor depends upon the modulation type. In this example, the modulation is BPSK and the phasors are real only, +1 or –1. Assuming the data sequence is {0011} bits, the data-phasor sequence is {𝑑𝑛 }={–1, –1, +1, +1}. The IDFT outputs are 𝑁−1 3 2𝜋 𝑗( 𝑛)×0 𝑥 (0) = ∑ 𝑑𝑛 𝑒 8 = ∑ 𝑑𝑛 = −1 − 1 + 1 + 1 = 0 𝑛=0 0 13  𝑁−1 𝑁−1 2𝜋 𝜋 𝑗( 𝑛)×1 𝑗( 𝑛 ) 𝑥 (1) = ∑ 𝑑𝑛 𝑒 8 = ∑ 𝑑𝑛 𝑒 4 𝑛=0 𝑛=0 𝜋 𝜋 𝜋 𝜋 𝑗( ×0) 𝑗( ×1) 𝑗( ×2) 𝑗( ×3) = (−1)𝑒 + (−1)𝑒 4 4 + (1)𝑒 4 + (1)𝑒 4 1 1 = −1 − (1 + 𝑗) + 𝑗 + (−1 + 𝑗) = (−√2 − 1) + 𝑗 = −2.41 + 𝑗 √2 √2 𝑁−1 𝑁−1 2𝜋 𝜋 𝑗( 𝑛)×2 𝑗( 𝑛 ) 𝑥 (2) = ∑ 𝑑𝑛 𝑒 8 = ∑ 𝑑𝑛 𝑒 2 = −1 + (−1)(𝑗) + 1 × (−1) + 1(−𝑗) = −2 − 2𝑗 𝑛=0 𝑛=0 𝑁−1 𝑁−1 2𝜋 3𝜋 𝑗( 𝑛)×3 𝑗( 𝑛 ) 𝑥 (3) = ∑ 𝑑𝑛 𝑒 8 = ∑ 𝑑𝑛 𝑒 4 𝑛=0 𝑛=0 1 1 = −1 + (−1)(−1 + 𝑗) + 1 × (−𝑗) + (1 + 𝑗) = 0.41 − 𝑗 √2 √2 2𝜋 𝑁−1 𝑗( 8 𝑛)×4 𝑁−1 𝑗 (𝑛𝜋) 𝑥 (4) = ∑𝑛=0 𝑑𝑛 𝑒 = ∑𝑛=0 𝑑𝑛 𝑒 = −1 + 1 + 1 − 1 = 0 The real part of 𝑥(𝑘) versus k (containing 8 samples) is plotted in the bottom-right portion of the figure (interpolated to a continuous waveform by the DACs). Figure 8 OFDM waveform-synthesis example: Zooming in on the IDFT. 14  To obtain the original data, the OFDM receiver must reverse the signal processing (using DFT instead of IDFT) that took place at the transmitter. Note the difference between 𝑁𝑐 and 𝑁. The parameter 𝑁𝑐 represents the number of data constellation points or data-occupied subcarriers being sent within an OFDM symbol, and the parameter 𝑁 represents the size (N-point) of the IDFT transform. For building realizable filters, the null bins, sometimes called zero extensions or zero padding, are used to form the IDFT transform to ensure that 𝑁 > 𝑁𝑐. It is interesting to note that in this example, we probe the channel with only positive frequencies (and DC) because it helps to easily observe the gated superposition of such familiar harmonics. Using only positive-occupied frequencies (asymmetric spectrum) requires complex time signals. In practice, we use both positive and negative frequencies (not DC) to send data independently on any tone in the available two sided OFDM spectrum. 6. Applications of OFDM 6.1 An Early OFDM Application: Wi-Fi Standard 802.11a Figure 9 summarizes the Wi-Fi 802.11a standard. modulation bandwidth 16 MHz, 4 MHz guard band, Therefore, a channel spacing of 20 MHz. Types of permitted data modulation: BPSK, QPSK, 16-QAM, or 64-QAM. The data rate to be one of eight choices: 6,9,12,18,24,36, 48,54 Mbps. The code rates can be 1/2, 2/3, or 3/4. The figure shows computations for the smallest and largest data rates. The number of subcarriers is 𝑁𝐶 = 52 (including 48 data tones and 4 pilot tones). The OFDM symbol formed at the IDFT output has: o a duration of 𝑇𝑆 = 3.2 μsec (data portion) o cyclic prefix 𝑇𝐶𝑃 = 0.8 μsec (CP portion) o OFDM symbol with time duration 𝑇𝑂𝐹𝐷𝑀 = 4.0 μsec. Thus, the symbol rate appears to be = 1/4μsec = 250 OFDM ksymbols/sec. Seeing this symbol rate, one might mistakenly be prompted to choose 250 kHz as the OFDM bandwidth, but that would not be correct because each OFDM symbol is made up of 𝑁𝐶 = 52 subcarriers spaced Δ f apart. The computation for OFDM bandwidth resembles that of ordinary M-FSK bandwidth. Since the data portion has duration 𝑇𝑆 = 3.2 μsec, the tone spacing Δ f between subcarriers is 1 𝛥𝑓 = = 312.5𝑘𝐻𝑧 (13) 𝑇𝑠 The OFDM modulation bandwidth can be calculated as 𝑊𝑂𝐹𝐷𝑀 = 𝑁𝐶 𝛥𝑓 = 16.250 ≈ 16 MHz (14) The maximum data rate can be given by 𝑅𝑏 = 𝑐𝑜𝑑𝑒 𝑟𝑎𝑡𝑒 × 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑑𝑎𝑡𝑎 𝑠𝑢𝑏𝑐𝑎𝑟𝑟𝑖𝑒𝑟𝑠 × 𝑠𝑦𝑚𝑏𝑜𝑙 𝑟𝑎𝑡𝑒 × 𝑚𝑜𝑑𝑢𝑙𝑎𝑡𝑖𝑜𝑛 𝑜𝑟𝑑𝑒𝑟 15  = 𝑟 × 𝑁𝑑 × 𝑅𝑆 × 𝑚 (15) For example, the minimum data rate is obtained assuming BPSK (m=1) and code rate 1/2. Then, 𝑅𝑏 = 𝑟 × 𝑁𝑑 × 𝑅𝑆 × 𝑚 = 0.5 × 48 × 250 × 103 × 1 = 6𝑀𝑏𝑝𝑠. The maximum data rate is obtained assuming 64-QAM (m=6) and code rate 3/4 as 𝑅𝑏 = 𝑟 × 𝑁𝑑 × 𝑅𝑆 × 𝑚 = 0.75 × 48 × 250 × 103 × 6 = 54 𝑀𝑏𝑝𝑠 Since in 802.11a, the data portion of each symbol has a duration of 𝑇𝑠 = 3.2 μsec, then for this example, we can see that the subcarrier tone spacing Δ f = 312.5 kHz. The OFDM sample rate 𝑓𝑠 can be similarly described (but controlled by the transform size N rather than 𝑁𝑐 ). Figure 9 802.11a OFDM application and examples. 1 𝑇𝑠 1 𝑇𝑂𝐹𝐷𝑀 The sampling interval 𝑇𝐿 = = = = 𝑓𝑠 𝑁 𝑁𝛥𝑓 𝑁𝐿 When N = 64, we can write 𝑓𝑠 = 𝛥𝑓 × 𝑁 ≈ 20 MHz (16) The example solution starts with a definition of the various parameters involving sample quantities. The counting parameters represent the number of subcarriers 𝑁𝑐 , transform size N, number of samples of cyclic prefix 𝑁𝑐𝑝 , and total number of samples per OFDM symbol 𝑁𝐿. 16  It is interesting that the CP does not enter into the OFDM bandwidth computation. Note that the subcarrier spacing ensures 50% spectral overlap between neighboring tones. Sometimes, for simplicity when describing OFDM, we show the N-point transform to be equal to the number of subcarriers 𝑁𝑐. But for realizable systems, the filtering dictates that 𝑁 > 𝑁𝑐. By using Equation (16), with the transform size N = 64, results in a sample rate 𝑓𝑠 of 20Msamples/s or 20 MHz. By doubling the transform size to N = 128, the sample rate doubles to 40 MHz. Increasing the transform size, and thus the sample rate has the effect of moving the spectral copies further apart. Figure 10 shows the two filter types that stem from a 20 MHz sampling rate in Figure 10a and a 40 MHz sampling rate in Figure 10b. (Note that the actual plotted transform values of 64 and 128 have been reduced to 20 and 40, respectively, for clarity.) In Figure 10a, where N = 64 in 802.11a (reduced to 20 for clarity), the transition bandwidth is quite narrow (steep transitions). A key benefit of a higher sampling rate is a larger separation between spectral replicates and, thus, an easier, less costly analog filter (following the DAC). Figure 10 OFDM signal filters in 802.11a are a function of transform size. (a) 20 MHz sample rate. (b) 40 MHz sample rate. Example 4 Typical OFDM Parameters for an 802.11a Local Area Network Summarize the important OFDM parameters and relationships in the 802.11a standard (displayed earlier in Figure 9). For computing parameter values, choose the transform size to be 64, and the sample rate (channel spacing) to be 20 MHz. 17  Start with the four different counting parameters: number of subcarriers, transform size, number of samples in the cyclic prefix, and total samples per OFDM symbol. Continue by showing sample time, time of data portion of OFDM symbol, time of OFDM symbol, symbol rate, and spacing between subchannels. Also verify the sample rate of 20 MHz, and compute the modulation bandwidth. Solution 𝑁𝑐 no. of subcarriers = 52 N = transform size (𝑁 > 𝑁𝑐 ) = 64 (or 128) 𝑁𝑐𝑝 = no. of CP samples (25% of N) = 16 𝑁𝐿 = total samples per OFDM symbol = N + 𝑁𝑐𝑝 = 80 𝑓𝑆 = sample rate (channel spacing or DFT bandwidth) = 20 MHz 𝑇𝐿 = sample time duration = 1 / 𝑓𝑆 = 1 / 20 MHz = 0.05 μs 𝑇𝑆 = data portion of OFDM symbol = N × 𝑇𝐿 = 64 × 0.05 μs = 3.2 μs 𝑇𝑐𝑝 = CP time duration = 𝑁𝑐𝑝 × 𝑇𝐿 = 16 × 0.05 μs = 0.8 μs 𝑇𝑂𝐹𝐷𝑀 = OFDM symbol time duration = 𝑇𝑆 + 𝑇𝑐𝑝 = 3.2 μs + 0.8 μs = 4 μs 𝑅𝑂𝐹𝐷𝑀 = OFDM symbol rate = 1 / 𝑇𝑂𝐹𝐷𝑀 = 1 / 4 μs = 250 ksamp / s Δ f = spacing between subcarriers = 1 / Ts = 1 / 3.2 μs = 312.5 kHz 𝑓𝑆 = sample rate (check on input value) = N × Δ f = 64 × 312.5 kHz = 20 MHz 𝑊𝑚𝑜𝑑 = modulation bandwidth = 𝑁𝑐 × Δ f = 52 × 312.5 kHz ≈ 16 MHz Example 5 Key Relationships Consider sample rate 𝑓𝑠 , data-symbol length 𝑇𝑠 for an OFDM system, we have locked in the required transform size N and subcarrier spacing Δ f. Describe the effects of (a) increasing the number N of the N-point IDFT and (b) decreasing Δ f frequency spacing between tones. Solution (a) Since 𝑓𝑠 = Δ f × N, an increase of N alone gives rise to an increase in sample rate but not bandwidth. A faster sample rate results in smaller 𝑇𝐿 or greater time-domain resolution and spreads the spectral copies further apart, making analog filtering easier and less costly. (b) Since 𝑇𝑠 = 1/Δ f, a decrease in Δ f alone increases the length of 𝑇𝑠 , making it less vulnerable (more robust) in withstanding a given channel ISI. 6.2 Long-Term Evolution (LTE) Long-Term Evolution (LTE) and LTE-Advanced are the standards, initiated in 2004, to exceed the requirements of the International Telecommunication Union (ITU) for a fourth generation (4G) radio communication. As of June 2014, LTE had become the fastest-growing cellular technology ever. The main attributes that distinguish LTE-Advanced are higher capacity with single- channel peak data rates of up to 1Gbps on the downlink and 500 Mbps on the uplink, latency > 𝑇𝑂𝐹𝐷𝑀 𝑇𝑂 𝑊𝑥𝑚𝑡 1 𝑓0 > > 𝑁𝐶 𝑇𝑂 𝑊𝑥𝑚𝑡 < 𝑁𝐶 < 𝑊𝑥𝑚𝑡 𝑇𝑂 𝑓0 1 1 𝑓0 ≈ = 4 = 250 kHz 5𝜎𝜏 5×(5)×10−6 𝑊𝑥𝑚𝑡 1.4 MHz 𝑁𝐶 𝑚𝑖𝑛 = = = 6 (Rounded up) 𝑓0 250 kHz 𝑁𝐶 𝑚𝑎𝑥 = 𝑊𝑥𝑚𝑡 𝑇𝑂 = 1.4 × 10−6 × 6.5 × 10−6 = 9 (Rounded down) Choose 𝑁𝐶 = 7 subcarriers, 𝑁 = 16 samples, 𝑁𝐶𝑃 = 4 samples 𝑁𝐿 = 20 samples Given R = 5.6 Mbps and 16-QAM. 𝑅 5.6×106 Then, 𝑊𝑥𝑚𝑡 = = = 1.4 𝑀𝑠𝑦𝑚𝑏𝑜𝑙/𝑠 𝑙𝑜𝑔2 𝑀 4 𝑊𝑥𝑚𝑡 1.4×106 𝛥𝑓 = = = 200 𝑘𝐻𝑧 𝑁𝐶 7 1 𝑇𝑂𝐹𝐷𝑀−𝑑𝑎𝑡𝑎 = = 5 𝜇𝑠𝑒𝑐 𝛥𝑓 1 𝑇𝐶𝑃 = (5 𝜇 𝑠𝑒𝑐) = 1.25 𝜇 𝑠𝑒𝑐 4 𝑇𝑂𝐹𝐷𝑀 = 𝑇𝑂𝐹𝐷𝑀−𝑑𝑎𝑡𝑎 + 𝑇𝐶𝑃 = 5𝜇 𝑠𝑒𝑐 + 1.25 𝜇 𝑠𝑒𝑐 = 6.25 𝜇 𝑠𝑒𝑐 𝑇 6.25 μ sec 𝑇𝑠𝑎𝑚𝑝𝑙𝑒 = = 𝑂𝐹𝐷𝑀 = 0.3125 μ sec 𝑁 20 𝐿 1 𝑓𝑆 = = 3.2 Msample/s 𝑇𝑠𝑎𝑚𝑝𝑙𝑒 1 𝑅𝑂𝐹𝐷𝑀 = = 160 𝑘𝐻𝑧 which is < 𝑓0. 𝑇𝑂𝐹𝐷𝑀 We can verify that the flat fading condition prevails by noting that the CP duration (1.25 μsec) is 64% longer than the given rms delay spread (0.8 μsec). We can verify that the slow 22 fading condition prevails by noting that the given coherence time (𝑇𝑂 = 6.5 μsec) is longer than the symbol time (𝑇𝑂𝐹𝐷𝑀 = 6.25 μsec). 7. Drawbacks of OFDM There are two main drawbacks to OFDM: o Frequency offset and Doppler shift between the transmitter and receiver cause intercarrier interference (ICI) in the DFT frame. o There is a relatively large peak-to-average power ratio (PAPR) associated with OFDM. 7.1 Sensitivity to Doppler Vehicular speeds are relatively small, at a maximum of around 100 miles/hr (approx 161 km/hr) for a vehicle and 300 km/hr for a high-speed train. At 5 GHz, these would translate to Doppler spreads of about 750 and 1400 Hz, which appear as a carrier-frequency offset that can be measured, estimated, and removed during the acquisition phase. Much higher speeds might cause the receiver-matched filter to become decorrelated, which would degrade orthogonality of the received signal set. For this reason, vehicles in LTE have access to lower throughputs than pedestrians. 7.2 Peak-to-Average Power Ratio (PAPR) The output of IDFT is the sum of gated sinusoids The summation of such signaling (due to different subcarrier frequencies and random data) yields a variety of amplitudes. Figure 14 shows how large peak-to-average power ratio (PAPR) is formed. Figure 14 PAPR problem in OFDM. 23  Single-carrier OFDM (SC-OFDM) is a creative approach for mitigating the PAPR problem. This approach is a modified version of OFDM that provides trade-off between living with high PAPR to achieve simple channel equalization and reducing PAPR. Motivation for Reducing PAPR Power amplifiers are DC-to-AC converters, and power pulled from the DC power supply (approximately constant) that is not delivered to the load is dissipated in the power amplifier. Amplifiers are very inefficient in their transduction process of turning DC power to signal power when they operate at small fractions of their peak power level. Typical efficiencies for an amplifier operating with an 802.11a signal are on the order of 18%. For Gaussian channels (Rayleigh-distributed envelope), the peak amplitude is about 4 times the average amplitude (and peak power is about 16 times the average power). Thus, an amplifier required to supply 1 Watt would have a peak power capability of 16 Watts and would be pulling 5.5 Watts from the power supply, squandering 4.5 Watts (raising the temperature of its heat sinks) while delivering 1 Watt to its external load. 8. Single-Carrier OFDM (SC-OFDM) SC-OFDM is a hybrid combination of the low PAPR characteristics of single-carrier transmission with the long symbol time of OFDM. In SC-OFDM, the data symbols first appear in the time domain as sequential Dirichlet functions [periodic extension of (sin x)/x due to sampling]. Then, a DFT process converts them to the frequency domain, where they are mapped to the desired location in the overall carrier bandwidth. Then, the IDFT operation converts them back to the time domain as Dirichlet functions, where the CP is added prior to transmission. The Dirichlet functions help to retain orthogonality, since the time signals are transmitted sequentially, each peaking at a different time. Another name for SC-OFDM is discrete Fourier transform spread OFDM (DFT- OFDM) Figure 15 provides a comparison of baseband transmitter outputs for OFDM sum of gated sinusoids (Figure 15a) and SC-OFDM sequential Dirichlet functions (Figure 15b). Figure 15b should make it clear that, although the data are random, the staggered data Dirichlet functions cannot add up unlike the case of the gated sinusoids. Thus their summation yields a much improved PAPR (no simultaneous peaks). Figure 16 is a simplified model of SC-OFDM symbol generation and reception. From this block diagram, we can see that many of the signal processing steps are common to both OFDM and SC-OFDM. The important difference is the added M-point DFT/IDFT steps in the single-carrier version. 24 Figure 15 Comparison of (a) sum of gated sinusoids with (b) sequentially transmitted Dirichlet functions. These are followed by the N-point IDFT/DFT steps similar to OFDM. Note that N > M. Also note that M is unrelated to the commonly used Mary modulation expression. Figure 16 Simplified model of SC-OFDM generation and reception. (Rumney, M.,“De-mystifying SC- FDMA, the New LTE Uplink” [white paper], Agilent Technologies, April 2008.) A comparison between the generation of OFDM and that of SCOFDM can be seen in the Figure 17 overviews. In Figure 17a, OFDM starts with a set of data points (constellation) in 2-space. 25  The analog sinc functions seen (not actually built) represent a visualization of sampled-data spectra, providing coefficients (amplitude and phase) for the OFDM sinusoidal basis set. Each data point modulates a separate subcarrier. The modulated subcarriers enter the input bins of the IDFT processor. Output samples form a time sequence that stems from the superposition of multiple gated sinusoids. Figure 17 (a) OFDM In Figure 17b, SC-OFDM also starts with a set of data points (constellation) in 2- space. But here, the analog sinc functions (which are not actually built) represent a visualization of a sampled-data time sequence, which is first operated on by the DFT. The DFT transforms the time sequence into spectral samples (magnitude and phase). Each data symbol, taken one at a time (with zero extensions), is transformed by the DFT to a wideband rectangular spectrum with a phase slope. Such spectra are time invariant over a data-symbol time. Next, the use of zero extensions into the IDFT results in an increased sample rate and better resolution via interpolation of the pulses out of the IDFT. Note that the subcarriers of each SC-OFDM output symbol are not the same across frequency but have their own fixed amplitude and phase for the SC-OFDM symbol duration. The sum of the time-invariant subcarriers represents a sequence of Dirichlet functions staggered in time, as shown in Figure 15. 26  Figure 28 (b) SC-OFDM. 19.1 SC-OFDM Signals Have Short Mainlobe Durations An SC-OFDM symbol is not constant over its time interval since it contains M-data symbols of a much shorter mainlobe duration: (1/M) times 𝑇𝑆. This is deceiving because each SC-OFDM time symbol is formed as a sum of constant-amplitude sinusoids that extend over the same long-OFDM symbol period. While the sinusoids cooperate so that their sum peaks over quite narrow time intervals, each SC-OFDM component (mainlobe and tails) extends over the same long-duration OFDM time interval. The key difference in the IDFT outputs for OFDM and SC-OFDM is that OFDM yields the superposition of gated sinusoids, and SC-OFDM yields staggered Dirichlet pulses. 27  Note that for OFDM, a simple sampling of the received waveform at the receiver does not reproduce the transmitted data, but for SC-OFDM, sampling the received waveform does reproduce the data. 9. Conclusion In this chapter, we have described OFDM, the creative multi-carrier signaling technique that is best characterized as a combination of modulation and multiplexing (or multiple access). The basic idea is to partition a high-data-rate signal into several low-data-rate signals, which can better survive the worst-case effects of multipath fading. We have illustrated applications of OFDM in standards such as 802.11 and Long- Term Evolution (LTE). The low-data-rate partitions yield lengthened symbol times. We have shown how multipath mitigation stems from the formation of such a longer orthogonal symbol together with a cyclic prefix (CP). Maintaining orthogonality is the key to improved OFDM performance in a fading environment. We have pointed out that one of OFDM’s drawbacks, called peak-to-average power ratio (PAPR) can be overcome with a special variant of OFDM, known as single- carrier OFDM (SC-OFDM). Whereas the basis functions for OFDM formation are gated sinusoids, for SC-OFDM they are staggered Dirichlet functions. ============================================================================= 28 10.Summarizing OFDM Waveform Synthesis The formation of an OFDM signal starts by modulating a binary data sequence into a parallel group of 𝑁𝑐 phasors. Each phasor is a complex number (I + jQ) obtained from the constellation diagram of the used modulation type such as BPSK, M-QAM,and M-PSK. It should be noted that in BPSK, the data constellation points are real, -1 and +1. The IDFT maps each of these modulation data points to sampled data subcarriers (gated sinusoids). Thus, the inputs to the IDFT can be visualized as narrow-bandwidth (sin x)/x spectral functions centered at frequency 𝑛𝑖 with spectral coefficient 𝐹(𝑖). Figure 6a provides a visualization of two narrow-bandwidth (sin x)/x spectral functions 𝐹(1)𝐻1 (𝑓– 𝑛1 ) and 𝐹(2)𝐻2 (𝑓– 𝑛2 ) inputs to the IDFT. These sketches are merely visualizations to help the reader verify the Fourier transform relationship between input and output. The IDFT forms a summation of these gated sinusoids, and the output consists of 𝑁 equally-spaced samples of that summation. Figure 6 Summary of OFDM waveform formation at the IDFT input and at the IDFT output. (a) IDFT input: Constellation points mapped into frequency bins. (b) IDFT output: Constellation points transformed into gated sinusoids. 29  Similarly, Figure 6b provides a visualization of the set of gated sinusoidal subcarriers out of the IDFT labeled real and imaginary ℎ1 (𝑡) and ℎ2 (𝑡), (where ℎ2 (𝑡) is the higher-frequency subcarrier). Each sinusoidal sequence has an integer number of cycles (0, 1, 2,…) per interval of length N. This is the important feature (contributing to orthogonality) that gives rise to the sinusoid’s last sample (N – 1) being adjacent to its first sample (0). It is useful to think of these sample points as plotted on a unit circle, such that the sample value of each sinusoid for time index N is the same as the sample value for time index 0. This end-around continuity holds true for the sum of these subcarriers. Thus, it holds true for the set of samples comprising an OFDM symbol. While each output-vector component is a summation of sampled-data sinusoids, there is no place in the IDFT process where one can actually “see” a sampled-data sinusoid. Keep in mind that the properties described here are preserved when the sinusoids and their transforms are sampled by the DFT and IDFT. Where are the continuous versions of the time series and spectra formed? The continuous versions of the time series and spectra are formed at the output of the digital-to-analog converter (DAC) shown in Figure 5 and launched by the output stage of the OFDM transmitter to be collected by the OFDM receiver. Continuous waveforms are not actually built because the IDFT and the DFT are discrete transforms (with discrete samples going in and discrete samples coming out). Figure 7 shows Time-Frequency Relation in OFDM There are three OFDM symbols located on a time axis. Each symbol occupying the time interval 𝑇𝑂𝐹𝐷𝑀 is made up of 𝑁𝐶 = 7 subcarriers, portrayed along a frequency axis. At each subcarrier location, we see a data symbol characterized as a two-dimensional constellation space. Two-space data modulation types, such as M-QAM, are the most suitable for OFDM. In this example, 16-QAM has been selected. Each point in this 16-ary constellation is a complex number, representing the amplitude and phase of a (sin x)/x shaped spectrum of the particular sinusoidal subcarrier (that the data constellation point has been coupled with). As shown in the figure, the spectral region over all the subcarriers is the OFDM’s transmission bandwidth, typically described as a two-sided spectrum. For the sake of the figure’s clarity, the 50% subcarrier overlaps are not shown. Note that the guard intervals (between OFDM symbols) mitigate the ISI. In this Figure 10, the kth sample of each OFDM symbol stems from the superposition of seven samples, one from each of its seven gated sinusoids at time t = k. In this figure, the F(0) subcarrier is at DC. In real-world OFDM systems, the DC subcarrier is never modulated with any data. 30 11.Hermitian Symmetry If a time signal is purely real (or purely imaginary), then its spectrum is complex and has Hermitian (or conjugate) symmetry and vice versa. For purely real signals, Hermitian symmetry means that the real part of the signal’s spectrum has even symmetry, and the imaginary part has odd symmetry. For purely imaginary signals, it’s just the opposite: The real part of an imaginary signal’s spectrum has odd symmetry, and the imaginary part has even symmetry. Hermitian properties can be examined in either the time or frequency domain, but their greatest usefulness is in the frequency domain. Figure 11 (left side) illustrates two spectra to be added: (1) a real time signal’s spectrum (real part with even symmetry, imaginary part with odd symmetry), and (2) an imaginary time signal’s spectrum (real part with odd symmetry, imaginary part with even symmetry). The right side of the figure shows a complex spectrum representing the typical sum of such spectra. It should be obvious that the complex spectrum (resulting from this summation) has no symmetry at all. If the two-sided spectrum has no symmetry, it means that its underlying time signal is not purely real and not purely imaginary; it is complex. Thus, Asymmetric spectrum can be decomposed to two Hermitian symmetric components that correspond to pure real and pure imaginary signals. Figure 11 Non-Hermitian spectrum of a complex baseband signal stemming from the summation of the spectra of real and imaginary time signals. Complex modulation gives us the ability to place independent data on the sidebands around zero frequency. Examine the baseband spectrum shown in Figure 12a and verify that its time signal must be complex (no Hermitian symmetry) and thus needs two wires for transmission. 31  Once this baseband signal is modulated onto an RF carrier, we can recognize that the resulting spectrum (in Figure 12b) of the up- and down-shifted baseband time signal (the transmission waveform) is Hermitian symmetric around zero. Hence, the RF time signal is real and can be transmitted over a single cable or antenna. Note that the real RF waveform whose spectrum is shown in Figure 12b is simply the real part of the upshifted baseband waveform. Any such real waveform is easily computed by adding to the up-shifted waveform its complex conjugate and dividing by 2, as follows: Baseband waveform: 𝑥(𝑡) + 𝑗𝑦(𝑡) Upshifted waveform: [𝑥(𝑡) + 𝑗𝑦(𝑡)] 𝑒 (𝑗𝜔𝑐𝑡) Real RF Waveform: 𝑅𝑒{[𝑥(𝑡) + 𝑗𝑦(𝑡)] exp(𝑗𝜔𝑐𝑡)} = 𝑅𝑒{[𝑥(𝑡) + 𝑗𝑦(𝑡)][𝑐𝑜𝑠 𝜔𝑐 𝑡 + 𝑗𝑠𝑖𝑛 𝜔𝑐 𝑡]} = 𝑥(𝑡) 𝑐𝑜𝑠 𝜔𝑐 𝑡 − 𝑦(𝑡) 𝑠𝑖𝑛 𝜔𝑐 𝑡 Figure 12 (a) Complex baseband and (b) real band-centered channels. 11.1 Properties of the Discrete Fourier Transform (DFT) In OFDM, the basis components in the transmitted time symbols {x(t)} are orthogonal. Each basis component has constant amplitude and is a gated sinusoid containing an integer number of cycles with abrupt discontinuities at its boundaries. During transmission through the channel, the continuous x(t) is linearly convolved with the channel’s impulse response ℎ𝐶 (𝑡) The channel interaction with the signal’s start and stop boundaries introduces continuous extended-length transients that alter the received signal in two ways: 32  Its length is increased, and its amplitude is reduced over the transient span. The first effect violates Rule 1 (preserve signal length), and the second effect violates Rule 2 (preserve constant envelope) for maintaining orthogonality. To ensure orthogonality, we must suppress the start/stop transients of the received waveform. Recall that circular convolution of two functions does not exhibit starting and stopping transients and does not increase the length of the resulting convolution. The way we suppress the transients of a linearly convolved time signal (with the channel impulse response) is to convert its convolution from linear to circular. The cyclic prefix (CP) is produced by copying a segment from the end of the time signal and is appended to the beginning of the signal. The back end of the original signal is of course continuous with its front end because the signal length 𝑇𝑆 matches the fundamental period (i.e., is a multiple of the period of its sinusoidal basis functions). Hence the back end of the appended CP is also continuous with the front end of the OFDM signal. There is no longer a transient at the original time signal’s starting edge; the transient now resides at the new starting edge of the cyclic prefix. If the CP end matches the signal front, as depicted in Figure 15, consider how this affects what happens during convolution. As the channel impulse response is sliding from the cyclic prefix into the signal interval, it has the appearance of leaving the signal’s back end and entering the front end. This makes the linear convolution appear to be circular. Such circular convolution is indeed what we would have if we didn’t still have the transient responses at the two boundaries of the signal. We complete the process at the receiver by discarding the CPs, which contain the starting and stopping transients of the received signal. After CP disposal, there are an integer number of cycles per symbol time–and in fact all three of the orthogonality rules are satisfied. Figure 15 Transmitted OFDM Symbols with Guard Intervals filled with Cyclic Prefix. The cyclic-prefix-end matches the signal-front. Recall that the transform of a sampled-data sequence is periodic. Thus, the convolution of its time series is circular, which nicely preserves the orthogonality of the waveforms. 33  In OFDM, the DFT of the received signal can be shown to be the multiplication X (n) and H(n) for each of the N subcarriers in the signal, a condition that is true only for circular convolution in the channel. What we have accomplished by appending the cyclic prefix and then discarding the two transients formed by linear convolution is to fool the system into performing circular convolution. 11.2 Linear and Circular Convolution Figure 16 indicates a message sequence 𝑥(𝑘) = 1, 2, 3, 2, 1 and an impulse response ℎ(𝑘) = 3, 2, 1. This convolution shows how the pulses sent from the transmitter are modified by the channel. The convolution can be obtained by obtained by aligning the time-reversed h (k) with x (k) in order to perform product summation, shift, and repeat until completion. In this example, the output sequence of this linear convolution is 3, 8, 14, 14, 10, 4, 1. Note that the transmitted sequence length was five symbols, but after linear convolution, it has increased to seven symbols.𝑀𝑜𝑢𝑡 = 𝑀𝑖𝑛𝑝𝑢𝑡 + 𝑀ℎ − 1 = 5 + 3 − 1=7 For shift m=0, For shift m=1, Figure 15 Linear convolution of data sequence and channel impulse response. What Would a Circularly Convolved x (k) ⊛ h (k) Look Like? 34  At the bottom of Figure 16, we see a circularized time axis with bold numbers representing bin addresses that establish the required alignment for circular convolution. Counterclockwise values on the outer circle represent positive bins, and clockwise values on the inner circle represent negative bins. Note that on the circularized axis, bin –2 corresponds to bin +3, and bin –1 corresponds to bin +4. Above this circle we see plotted h (k) and h (– k). We also see an h (– k) plot that has been made causal by showing each of its sample values placed in positive bins. Now we are ready to start convolution with circular indexes. 35  Figure 15.17 What would circular convolution look like? Figure 15.18 aligns the sampled data x(k) and the time-reversed impulse response h (– k) from the circularized time index, as needed, for the product summation and shift steps. The resulting circularly convolved output 7, 9, 14, 14, 10 in the figure is quite different from the linearly convolved output; an obvious difference is that sequence length has been preserved. Recall why we want the convolution to be circular rather than linear: Circular convolution helps us maintain orthogonality, as described in Sections 14.2 and 14.3. Circular convolution yields a very different looking output. 36  Figure 15.18 Circular convolution. The Trick That Makes Linear Convolution Appear Circular We add a CP (at least as long as the channel delay spread) so that the signal will look periodic. In the case of the message-symbol sequence x(k) = 1, 2, 3, 2, 1 introduced in Section 14.4, let us replicate the last two symbols and insert them at the start of the sequence, as the CP. In that way, the new sequence x′(k) can be written as x′(k) = 2, 1, 1, 2, 3, 2, 1. Next, we perform linear convolution with the channel impulse response h(k) = 3, 2, 1, just as we did in Figure 15. In this case, as shown in Figure 18b, the resulting sequence is 6, 7, 7, 9, 14, 14, 10, 4, 1. In Figure 18, we can now compare the circular convolution output x (k) ⊛ h (k) plotted in Figure 18a with the linear-convolution output x′(k) * h(k) plotted in Figure 18b. At the receiver, the CP portion (lengthened to four samples) of the linearly convolved sequence is discarded. The remaining five sample values 7, 9, 14, 14, 10 are exactly the same as those obtained with circular convolution of the original five-sample sequence. 12.The Cyclic Prefix (CP) and Orthogonality in OFDM The CP helps maintain orthogonality and converts the task of equalization into one of simple spectral scaling. To accomplish this, the CP modifies linear convolution (of the transmitted signal with the channel impulse response) so that it appears to be circular convolution. 12.1 Properties of Continuous and Discrete Fourier Transforms For continuous time signals, spectral multiplication 𝑋(𝑓)𝐻(𝑓) corresponds to linear convolution x (t) * h (t) in time. A property of the discrete Fourier transform (DFT) is that spectral multiplication of sampled signals 𝑋(𝑛)𝐻(𝑛) corresponds to circular convolution 𝑥 (𝑘) ⊛ ℎ(𝑘) in time. In IDFT, the transform is sampled which makes the time signal periodic. Thus, 𝑥 (𝑘) ⊛ ℎ(𝑘) is periodic in time. 37  Similarly, sampling the time signal makes the transform periodic. Figure 14 shows a DFT of a sampled rectangular function, plotted on a unit circle (with the same value at start and finish). This plot facilitates comparison of the start and finish, knowledge of any transients, and the confirmation of periodicity and steady state (no transients), when the start and finish values are identical. It should be noted that the process of sampling continuous time signals dictates that their spectra must be band limited. For continuous time waveforms, spectral sidelobes decay as 1/f as f approaches ∞. Once the spectrum decays below the noise, it is “essentially” band limited. Figure 14 The discrete (sin x/x) Fourier transform of a sampled rectangular function, using circular convolution, plotted on a unit circle. Frequency can be depicted as an angular coordinate on the circle. When using DFTs for implementing OFDM systems, a continuous transmitted waveform linearly convolved with the channel impulse response is modified so that it appears to be circularly convolved with the channel impulse response. This allows equalization to become a simple spectral scaling during the DFT process. 12.2 Reconstructing the OFDM Subcarriers For reconstructing the correct OFDM subcarriers at the receiver, the most important action needed is maintaining signal orthogonality among the RF subcarrier pulses (gated sinusoids). This is primarily accomplished by following three rules: o Rule 1: Preserve signal length o Rule 2: Preserve constant envelope o Rule 3: Preserve an integer number of cycles per gated sinusoid 38  Let us review these rules as well as the uniqueness of (gated) sinusoids. Preserving Signal Length The use of linear convolution with an N-point DFT would create a lengthened output. But by making the signal (with a CP) appear circularly convolved with the channel impulse response and then discarding the CP containing the channel transients, the original signal length is preserved. Preserving a Constant Envelope Convolving a signal with the channel impulse response causes a transient at the start and end of the symbol. Any such transient causes envelope variations. The CP absorbs the starting transient of the current symbol and the stopping transient of the previous symbol. By discarding the CP in the guard interval, where the overlapping transients reside, a constant signal envelope is thereby preserved for each gated sinusoid. Preserving an Integer Number of Cycles No matter how bad the channel might be, for a given frequency, the steady-state response (ssr) for a sinusoidal input is just an amplitude and phase variation of the sinusoid but remains sinusoidal with the same frequency. This is only true for steady-state responses; it is not the case for a time-varying channel. Discarding the CP guard interval preserves the integer number of cycles in each symbol (the way it was originally created). This happens because the preserved, steady-state response interval is designed to span an integer number of cycles. 39 Figure 18 19 Comparison of (a) circular and (b) linear convolution. 40