# Basic Principles of OFDM Notes

### 1. Introduction

The basic principle of OFDM is to split a high-rate datastream into a number of lower rate streams that are transmitted simultaneously over a number of subcarriers. Because the symbol duration increases for lower rate parallel subcarriers, the relative amount of dispersion in time caused by multipath delay spread is decreased. Intersymbol interference is eliminated almost completely by introducing a guard time in every OFDM symbol. In the guard time , the symbol is cyclically extended to avoid intercarrier interference. In OFDM design, a number of parameters are up for consideration, such as the number of subcarriers, guard time, symbol duration, subcarrier spacing, modulation type per subcarrier. The choice of parameters is influenced by system requirements such as available bandwidth, required bit rate, tolerable delay spread, and Doppler values. Some requirement are conflicting. For instance, to get a good delay spread tolerance, a large number of subcarriers with small subcarrier spacing is desirable, but the opposite is true for a good tolerance against Doppler spread and phase noise.

### 2. Data transmission using multiple carriers

An OFDM signal consists of a sum of subcarriers that are modulated by using phase shift keying (PSK) or qudrature amplitude modulation (QAM). If i d are the complex QAM symbol, s N is the number of subcarriers, T the symbol duration, and T f f i i = + 0 the carrier frequency, then one OFDM symbol starting at s t = t can be written as: __ −− = = − 1 0 ( ) Re{ exp( 2 ( )}, s N i i i s s t d j f t t t t t T s s £ £ + (1) s(t ) = 0, s t < t Ù t t T s > + In the literature, often the equivalent complex notation is used, which is given by (2). In this representation, the real and imaginary parts correspond to the in-phase and quadrature parts of the OFDM signal, which have to be multiplied by a cosine and sine of the desired carrier frequency to produce the final OFDM signal. Figure (1) shows the operation of the OFDM modular in block diagram. __ −− = = − 1 0 ( ) exp( 2 ( ), s N i i i s s t d j pf t t t t t T s s £ £ + (2) s(t ) = 0, s t < t Ù t t T s > + As an example , figure (2) shows four subcarriers from one OFDM signal. In this example, all subcarriersn have the phase and amplitude, but in practice the amplitudes and phases may be modulated differently for each subcarrier. Note that each subcarrier has exactly an integer number of cycles in the interval T , and the number of cycles between adjacent subcarries differs by exactly one. This properly accounts for the orthogoality between subcarriers. For instance, if the jth subcarrier from (2) is demodulated by downconverting the signal with a frequency of T j f f j = + 0 and then integrating the signal over T seconds, the result is as written in (3). By looking at the intermediate result, it can be seen that a complex carrier is integrated over T seconds. For the demodulated subcarrier j, this integration gives the desired output j d (multiplied by a constant factor T ), which is the QAM value for that particular subcarriers. For all other subcarriers , this integration is zero, because Parallel To Serial converter QAM data exp( 2 ( )) 0 s j pf t − t exp( 2 ( )) N 1 s j f t t s − − p OFDM signal OFDM modulator the frequency difference T (i − j) produce an integer number of cycles within the integration interval T ,such that the integration result is always zero.

_ _ _ _ − = + + − = − = − = − − − 1 0 1 0 exp( 2 ( )) exp( 2 ( )) exp( 2 ( )) s s s s s s N i t T t i s j t T t N i j s i i s t t dt d T T i j d j j f t t d j f t t dt p p p (3) The orthogonality of different OFDM subcarriers can also be demonstrated in another way. According to (1), each OFDM symbol contains subcarriers that are nonzero over a T -seconds interval. Hence, the spectrum of a single symbol is a convolution of group of Dirac pulses located at the subcarrier frequencies with the spectrum of a square pulse that is one for a Tsecond period and zero otherwise. The amplitude spectrum of the square pulse is equal to sinc(pfT) , which has zeros for all frequencies f that are an integer multiple of 1/T . This effect is shown in figure which shows the overlapping sinc spectra of individual subcarriers. At the maximum of each subcarrier spectrum, all other subcarrier spectra are zero. Because an OFDM receiver calculates the spectrum values at those points that correspond to the maxima of individual subcarrier, it can demodulate each subcarrier free from any interference from the other subcarriers. Basically, Figure (3) shows that the OFDM spectrum fulfills Nyquist’s criterion for an inter symbol interference free pulse shape. Notice that the pulse shape is present in frequency domain and note in the time domain, for which the Nyquist criterion usually is applied. Therefore, instead of intersymbol interference (ISI), it is intercarrier interference (ICI) that avoided by having the maximum of one subcarrier spectrum correspond to zero crossing of all the others.
3. Generation of subcarriers using the IFFT

The complex baseband OFDM signal as defined by (2) is in fact nothing more than the inverse Fourier transform of S N QAM input symbol. The time discrete equivalent is the inverse discrete Fourier transform (IDFT), which is given by :

__ −− = = 1 0 ( ) exp( 2 ) s N i i N in s n d j p (4) Where the time t is replaced by a sample number n. In practice, this transform can be implemented very efficiently by the inverse Fast Fourier transform (IFFT) as shown in figure(4) and (5).

QAM Modulator Serial to Parallel converter IFFT Parallel to Serial converter D/A s(t ) exp( 2 ) 0j pf t 0 d 1 d −1 Ns d RbpsTrransmiitttterr QAM demodulator Serial to Parallel converter FFT Parallel to Serial converter D/A s(t ) exp( 2 ) 0− j pf t 0 d 1 d −1 s N d LPS Rbps Receiiverr

### 4. Guard time and cyclic extension

One of the most important reasons to do OFDM is the efficient way it deals with multipath delay spread. By dividing the input data stream in s N subcarriers, the symbol duration is made s N times smaller, which also reduces the relative multipath delay spread, relative to symbol time, by the same factor. To eliminate intersymbol interference almost completely, a guard time is introduced for each OFDM symbol. The guard time is chosen larger than the expected delay spread, such that multipath components from one symbol cannot interfere with the next symbol. The guard time could consist of no signal at all. In that case, however, the problem of intercarrier (ICI) would arise. ICI is crosstalk between different subcarriers,which means they are no longer orthogonal. This effect is illustrated in figure (6) in this example, a subcarrier 1 and a delayed subcarrier 2 are shown. When an OFDM receiver tries to demodulate the first subcarrier, it will encounter some interference from the second subcarrier, because within the FFT interval, there is no integer number of cycles difference between subcarrier 1and 2. At the same time, there will be crosstalk from the first to the second subcarrier for the same reason.

To eliminate ICI, the OFDM symbol is cyclically extended in the guard time, as shown in figure(7). This ensures that delayed replicas of the OFDM symbol always have an integer number of cycles within the FFT interval, as long as the delay is smaller than the guard time. As result, multipath signals with delays smaller than the guard time cannot cause ICI.

As an example of how multipath effects OFDM, figure(8) shows received signal for tow-ray channel, where the dotted curve is a delayed replica of the solid curve. Three separate subcarriers are shown during three symbol intervals.

In reality, an OFDM receiver only sees the sum of all these signals, but showing the separate components makes it more clear what the effect of multipath is. From the fig that there can be 180-degree phase jumps at the symbol boundaries. For the dotted curve, these phase jumps occur at a certain delay after the first path. In this particular example, this multipath delay is smaller than the guard time, which means there are no phase transition during the FFT interval. Hence, an OFDM receiver “sees” the sum of pure sine waves with some phase offsets. This summation does not destroy the orthogonality between the subcarries, it only introduces a different phase shift for each subcarrier. The orthogonality does become lost if the multipath delay becomes larger than the guard time. In that case, the phase transitions of delayed path fall within the FFT interval of the receiver.

The summation of the sine waves of the first path with the phase modulated waves

of the delayed path no longer gives a set of orthogonal pure sine waves, resulting in a certain level of interference. To get an idea what level of interference is introduced when the multipath delay exceeds the guard time, Figure (9) depicts three constellation diagrams that were drived from a simulation of an OFDM link with 48 subcarriers, each modulated by using 16-QAM. Figure (9)a shows the undistorted 16-QAM constellation, which is observed whenever the multipath delay is below the guard time. In figure (9)b, the multipath delay exceeds the guard time by a small 3% fraction of the FFT interval. Hence, the subcarriers are not orthogonal any more but the interference is still small enough to get a reasonable received constellation. In The interference is now so large that the constellation is seriously blurred, causing an unacceptable error rate.

### 5. windowing

In the previous sections, it was explained how an OFDM symbol is formed by performing an IFFT and adding a cyclic extension. Looking at an example OFDM signal like in figure (8), sharp phase transition caused by the modulation can be seen at the symbol boundaries. Essentially, an OFDM signal like the one depicted in figure (8) consists of a number of unfiltered QAM subcarriers. As a

result, the out-of-band spectrum decreases rather slowly, according to a sinc function. As an example of this, the spectra for 16, 64, and 256 subcarriers are plotted in Figure (10). For larger number of subcarriers, the spectrum the spectrum goes down rapidly in the beginning, which is caused by the fact that the sidelobes are closer together. However, even the spectrum for 256 subcarriers has relatively large –40 dB bandwidth that is almost four times the – 3 dB bandwidth.

To make the spectrum goes down more rapidly, windowing can be applied to the individual OFDM symbol. Windowing an OFDM symbol makes the amplitude go smoothly to zero at the symbol boundaries. A commonly used window type is the raised cosine window. Here, s T is the symbol interval, which is shorter than the total symbol duration because we allow adjacent symbols to partially overlap in the roll-of region. In equation form, an OFDM symbol starting at time s s t = t = kT is defined as:

( ) 0, (1 ) (1 ) ( ) Re{ ( ) exp( 2 ( ))}, 1 0 b b p = < Ù > + + £ £ + + − − − = __ −− = k s s s s s s N i s i i s prefix s t t t t t T t t t T s t w t t d j f t t T s where b is the roll-off factor of the raised cosine

In practice, the OFDM signal is generated as follows: first, s N input QAM values are padded with zeros to get N input samples that are used to calculate an IFFT. Then, the last perfix T samples of the IFFT output are inserted at the start of the OFDM symbol, and the first postfix T samples are appended at the end. The OFDM symbol is then multiplied by a raised cosine window w(t) to more quickly reduce the power of out-of-band subcarriers. The OFDM symbol is then added to the output of the previous OFDM symbol with a delay of s T , such that there is an overlap region of bTs , where b is the roll-off factor of the raised cosine window. Instead of windowing, it is also possible to use convolutional filtering techniques to reduce the effect of the out-of-band spectrum.

### 6. Choice of OFDM parameters

The choice of various OFDM parameters is a tradeoff between various, often conflicting requirements. Usually, there are three main requirements to start with: bandwidth, bit rate, and delay spread. The delay spread directly dictates the guard time. As a rule, the guard time should be about two to four times the root-mean-squared delay spread. This value depends on the type of coding and QAM modulation. Higher order QAM (like 64-QAM) is more sensitive to ICI and ISI than QPSK; while heavier coding obviously reduces the sensitivity to such interference.

Now the guard time has been set, the symbol duration can be fixed. To minimize the signal-to-noise ratio (SNR) loss caused by guard time, it is desirable to have the symbol duration much larger than the guard time. It cannot be arbitrarily large, however, because a larger symbol duration means more subcarriers with a smaller subcarrier spacing, a larger implementation complexity, and more sensitivity to phase noise and frequency offset, as well as an increased peak-to-average power ratio. Hence, a practical design choice to make the symbol duration at least five times the guard time, which implies a 1dB SNR loss because the guard time. After the symbol duration and guard time are fixed, the number of subcarriers follows directly as the required –3 dB bandwidth divided by the subcarrier spacing,which is the inverse of the symbol duration less the guard time. Alternatively, the number of subcarriers may be determined by the required bit rate divided by the bit rate per subcarrier. The bit rate per subcarrier is defined by the modulation type, coding rate, and symbol rate. An additional requirement that can affect the chosen parameters is the demand for an integer number of samples both within the FFT/IFFT interval and in the symbol interval.