OFDM Understanding (reprint)

Description: The following text, gray for blowing hydrology, black for the body, blue for the use of the parameters in the actual application of the instructions.

The cause is this. Time back to the end of 07, 4G Square Hing, next door next door "small white" classmate said see not quite understand ofdma principle, let me explain. I have always been very confident in my skill level, logical thinking and expression skills, so I agreed with a chuckle. Half an hour later, in the attempt to explain from the time domain, frequency domain and physical meaning, but can not from the "small white" eyes to erase the confusion, I put up the white flag, let "small Bai" to fend for themselves.

The mastery of knowledge and ability, my own rough divided into two layers: the first layer is "will, can apply", the second layer is "understand, can derive". And can be explained, and let people understand, is probably the distinction between a layer and two levels of the watershed. To make a cock-and-thread analogy: The first layer is the cultivation of the people, even if the "will", but also has built a base, elixir, dollar and other realms of the points, and the entrance examination is the days of robbery, not to the Mahayana state, eventually to become a robber, the second layer is heaven, also has its own fairy, Jinxian of the points, and can be repaired to the Tao I always feel that I am a professional is a "little fairy", but it is "small white" hit the face.

This is a big negative impact on me, one is the suspicion of their own technology home for a long time, the ability to express serious degradation "for example, I occasionally search for a precise verb or adjective, need to try 2-3 times, or more"; second, when it comes to OFDM content, As if to see a white paper on the prowling of a lingering black flies.

After many years, and recently reviewed the OFDM, inadvertently and remember this case, hesitated repeatedly, or decided to take the time to write this article, the only hovering in the brain of the "black flies" shot dead. Therefore, although the network resources are very rich, various articles can be searched, in fact, I do not need to write this article is not necessarily better than others. But after all, it is the lack of their own, need to fill up.

The following attempts to illustrate the main interpretation of OFDM, with "easy to understand" as the first. "Little white", are you ready?

Note: The following discussion, if not described, is assumed to be an ideal channel.

Chapter One: OFDM on the time domain

The "O" of OFDM stands for "quadrature", so let's talk about orthogonal.

First of all, in the simplest case, sin (t) and sin (2t) are orthogonal "proofs: sin (t) sin (2t) has an integral of 0" on the interval [0,2π], and the sine function is the most visual description of the wave, so we use this as the intervention point. Since this article is about illustrations, let's use a graphical approach to understand orthogonality first. "If you can look at this problem from the perspective of vector space, you are not" white ", R U?" strategically advantageous position.

In the following illustration, the signal is transmitted in the time of [0,2π] with the most understandable amplitude modulation: sin (t) transmits signal A, so sends a sin (t), sin (2t) transmits signal B, so it sends B sin (2t). The use of sin (t) and sin (2t) is used to carry the signal, is to receive the pre-specified information, in this article is called the sub-carrier, modulation on the sub-carrier amplitude signals A and B, is the message to be sent . Therefore, the signal transmitted in the channel is a sin (t) +b sin (2t). At the receiving end, A and b are obtained by pairing the received signal with the integral detection of sin (t) and sin (2t) respectively. (in shape with Google draw)

Figure A: The sin (t) that sends a signal

Figure two: sin (2t) sending B-Signal "NOTE: two full waveforms were sent within the interval [0,2π]"

Figure three: The superimposed signal sent in the wireless space a sin (t) +b sin (2t)

Figure Four: The received signal is multiplied by sin (t), and the integral decodes a signal. "The Sin (2t), which transmits the B-signal, is 0" after the integral, as described above

Figure V: The received signal is multiplied by sin (2t), and the integral decodes the B signal. "As mentioned above, the sin (t) of transmitting a signal, after the integral is 0"

Figure VI: Flowchart

When you get here, you may have two states:

One is: Ah, the original is so, I understand.

One is: Ah, how can this, I completely can not imagine. The point here is that you don't have to imagine (visualize). The mathematics is so defined orthogonal, the mathematical proof of their orthogonality, then they are orthogonal, "they can not interfere with each other to carry their own information." Choose sin (t) and sin (2t) as an example, and formally because they are somewhere between the intuitive and abstract transitions, go over it.

The above diagram is simple, but it is all complex.

1.1 Next, Extend sin (t) and sin (2t) to more sub-carrier sequences {sin (2π δf t), sin (2π δf 2t), sin (2π δf 3t),..., sin (2π δf kt)} (such as k=16,256,1024, etc. ), should be a good understanding of things. Among them, 2π is a constant; Δf is a pre-selected carrier interval and a constant. 1T,2T,3T,..., KT guarantees the orthogonality of the sine wave sequence.

1.2 Next, the cos (t) is also introduced. It is easy to prove that cos (t) is orthogonal to sin (t) and is orthogonal to the entire sin (kt) orthogonal family. Similarly, the Cos (KT) is also orthogonal to the entire sin (kt) orthogonal family. So the launch sequence expands to {sin (2π δf t), sin (2π δf 2t), sin (2π δf 3t),..., sin (2π δf kt), cos (2π δf t), cos (2π δf 2t), cos (2π δf 3t), ..., cos (2π δf kt)} is also a logical.

1.3 After the first two steps of expansion, selected 2 sets of orthogonal sequence sin (kt) and cos (KT), this is only the transmission of "media." The actual information to be transmitted also needs to be modulated on these carriers, namely sin (t), sin (2t),..., sin (kt) respectively amplitude modulated A1,A2,..., ak signal, cos (t), cos (2t),..., cos (kt) respectively amplitude modulated b1,b2,..., BK Signal. This 2n group of orthogonal signals sent out at the same time, the space will be superimposed on what kind of waveform? Do a simple addition as follows:

F (t) = A1 sin (2π δf t) +
A2 sin (2π δf 2t) +
A3 sin (2π δf 3t) +
...
AK sin (2π δf kt) +
B1 sin (2π δf t) +
B2 sin (2π δf 2t) +
B3 sin (2π δf 3t) +
...
BK sin (2π δf kt) +
=∑ak sin (2π δf kt) +∑bk cos (2π δf kt) "equation 1-1: expression of real numbers"

In order to facilitate the mathematical processing, the above formula has the plural expression form as follows:
F (t) =∑fk E (J 2π δf kt) "equation 1-2: The expression of complex numbers, this editor can not find the upper corner superscript ... But you should be able to read it.

The above formula can look like this: Each sub-carrier sequence is sending its own signals, overlapping each other in the air, and finally the receiver to see the signal is f (t). After receiving the mixing signal f (t), and then on each sub-carrier after the multiplication of the integration of the operation, you can remove each sub-carrier signal is carried out separately.

Then, take a look at Equation 1-1 and equation 1-2!!! Find out? This is the Fourier series . If T is discretized, then it is a discrete Fourier transform . So there is the story that OFDM realizes by FFT. More descriptions will be made in the sections below.

According to the ancient tradition, F represents the frequency domain, F represents the time domain, so can be seen from Equation 1-2, the amplitude of the modulation above each sub-carrier, is the frequency domain information. A similar argument is that OFDM transmits a frequency domain signal. This is a bit awkward, but many tutorials or articles use this way of explaining how the reader understands it. If purely from the formula or sub-carrier view, this statement is actually very straightforward elaboration.

The above 1.1-1.3 extension, as shown:

Figure VII: The OFDM system diagram on the time domain


1.4 Is there another step? Actually, there is. "Small white" you can think first, the unexpected words first look down, because this needs to be considered in the frequency domain, so I wrote in the back. "also refer to [1]"

The above-mentioned time domain analysis with LTE implementation, the following conditions:
"NOTE 1: This paragraph describes the need to have the basic knowledge of the LTE physical layer, if you do not understand, please temporarily skip, read the whole article and then back to see"
"Note 2:lte is not a time domain implementation, the following is only the use of LTE parameters, to do a reference analysis"

Sub-carrier interval δf=15khz, the transmission time of an OFDM symbol is 66.7US, can be found, 15khz*66.67us=1, that is, the baseband on the transmission time of an OFDM symbol to send a full waveform of a single harmonic. For 10M LTE systems, 1024 sub-carriers are used, but only the middle 600 (without the most intermediate DC) subcarrier is used to transmit data. In the time of an OFDM symbol (i.e. 66.67us), close to the middle of the two harmonic transmission of a complete waveform, and then the outside point of two two harmonics transmission two full waveform, and so on to the outermost two 300 harmonics transmitted 300 complete waveforms. Within this 66.67us, 600 sub-carriers are orthogonal to each other and carry 600 complex signals on each of them.

The above statement is a bit verbose, not intuitive. Would have been ready to draw another picture, but one of the above already has a similar figure, the real is the same; second, 600 sub-carriers, too many points ...

OK, speaking of this, from the time domain above the OFDM, in fact, is quite simple and crisp pleasing. However, a system to realize OFDM from the time domain, the difficulty is too large, the delay and frequency offset will seriously destroy the sub-carrier orthogonality, thus affecting the performance of the system. This will be mentioned in various teaching materials, I will not repeat it.

The following will be transferred to the frequency domain to describe the OFDM, because the frequency domain is not very intuitive, it will be slightly confusing. But as long as you think about the overlap between the time domain subcarrier, everything will be fine.

Chapter Two: OFDM in frequency domain

The discussion at the time domain of the first chapter begins with the "O" in OFDM, and in the frequency domain of this chapter we start with "FDM".
First legend a system diagram of a conventional FDM:

Figure 11: Conventional FDM, where there is a gap between two signal spectrums, without interfering with each other

To make better use of the system bandwidth, the sub-carrier can be spaced as close as possible.

Figure 12: Close to FDM, in practice, considering the hardware implementation, demodulation of the first signal, it is difficult to completely remove the impact of the second signal (the implementation of the circuit after all, can not be as neat as scissors cutting paper), the two signals may have caused interference with each other

Can you come closer? OK. This is the origin of OFDM, ah, nearly equal to the Nyquist bandwidth (detailed later), so that the frequency band utilization reached the theoretical maximum value.

Figure 13: Continue closer, the interval frequency is orthogonal to each other, so the spectrum is overlapping, but there is still no interference with each other. The Magic of OFDM

Above three is a little bit of pediatrics, do not know "small white" is not already in the heart shout: This who do not know! But I've taken the time to draw three pictures here, and there's always something to consider:
A. As a supplement and a connection between the previous chapter and this chapter, the performance of OFDM in the frequency domain, i.e. the origin of OFDM.
B. Lead thinking: What is the bandwidth of the signal?
C. Guide thinking: How wide is the OFDM orthogonal spectrum overlay? Combined with 1.4, think first, then look down, it will be better.

Back on track again, please look at the first section of figure one to figure six time domain waveform diagram, shown in the time domain, waveform modulation, overlay reception, as well as the final decoding. Let's take a look at how each of the steps in figure one to figure three behaves in the frequency domain.

First Look at sin (t). "Small white" ah "small white", you say then say sin (t) spectrum is what ah? "Small white" weak weak said: is an impulse. Yes, sin (t) is a single sine wave that represents a single frequency, so its spectrum is naturally an impulse. However, the sin (t) shown in figure one is not really a sin (t), but only a small segment within [0,2π]. The infinite length of the signal is limited to a short period of time, "it is like catching a hair from a complete person, and then throw the whole person to the hair generation" the spectrum is no longer an impulse.

The sin (t) signal, which is limited to [0,2π], is the equivalent of an infinitely long sin (t) signal multiplied by a gate signal (rectangular pulse) on [0,2π], whose spectrum is the convolution of the spectrum. The Spectrum of sin (t) is impulse, and the spectrum of the gate signal is the sinc signal (that is, the sin (x)/X. Signal). The impulse signal convolution sinc signal is equivalent to the removal of the sinc signal. So the analysis here, can be obtained Tuyi time domain waveform of its corresponding spectrum is as follows:

Figure 21: Spectrum of the A sin (t) signal limited to [0,2π], i.e. the spectrum of modulated signals with a sin (t) carrier

The spectral analysis of sin (2t) is basically the same. It is important to note that because the orthogonal interval is [0,2π], sin (2t) sends two full waveforms at the same time. The same gate function ensures that the spectral shape of the two functions is the same, except where the spectrum is moved :

Figure 22: Spectrum of the B sin (2t) signal limited to [0,2π], i.e. the spectrum of modulated signals with a sin (2t) carrier

The spectrum of the signals transmitted by sin (t) and sin (2t) is superimposed, as follows:

Figure 23:a sin (t) +b sin (2t) signal spectrum


Figure 23 and Figure 13 are all spectra of two orthogonal sub-carriers in the frequency domain. A little bit, did you find out? Not quite the same!

Yes, you must have thought of it, because the baseband signal is typically passed through the pulse-shaped filter before transmission. For example, with the "L-cosine roll-down filter", the signal shown in Figure 23 will be repaired to the signal shown in Figure 13. This can effectively limit the external signal of the bandwidth, in the case of ensuring that the signal does not have crosstalk between the code, the maximum use of bandwidth, but also to reduce the sub-carrier signal interference between each channel. This is also not mentioned in 1.4, more reference [1]

Tip: The pulse-forming filter acts in the frequency domain, and each code element in the time domain can be "considered" to be emitted in a similar sinc signal. There is no need to dwell on the time domain waveform of the sending side code element, just need to know at the receiving end through the appropriate sampling can be no distortion of the recovery signal is OK.

Here is the nyquist first rule , which is described in the following box:

Nyquist First rule Please Google, here is the inference: The code element rate is 1/t (that is, each code element transmission time is T), in the case of cross-talk crosstalk transmission, the minimum bandwidth required is called the Nyquist bandwidth. For ideal low communication lanes, Nyquist bandwidth w = 1/(2T) For ideal band communication channel, the Nyquist bandwidth w = 1/t In Figure 31 below, you can see that the actual bandwidth B of the signal is greater than the Nyquist bandwidth W (Low-pass 1/(2T) or bandpass 1/t), which is the ideal and realistic distance. Supplementary note: The "bandwidth" mentioned in this article, also known as the conventional bandwidth understanding method, refers to the >=0 part of the signal spectrum. In the removal process from low to bandpass, because the negative frequency portion of the original signal is also moved out (also can be understood as the result of the same multiplication E (J2ΠFCT) + E (-J2ΠFCT), see reference [2]) "Note: there is no upper corner superscript and subscript editor, really uncomfortable. However, you should be able to read the ", so the bandwidth doubled." As shown in the following: Figure 31: A rich map, please refer to the above and below explanatory text

The first rule of Nyquist is outlined above, and one important purpose is to illustrate the issue of lower band utilization. The frequency band utilization is the ratio of the 1/t of the code element rate to the bandwidth B (or W) .

Ideally, the low-communication channel real-number signal, the frequency band utilization is 2baud/hz; With the communication channel complex, the frequency band utilization is also 2baud/hz (this is my way of understanding, the complex signal is equivalent to two real signal "complex number is only conducive to mathematical calculation, engineering implementation can only be real, that is, using sin (t And the orthogonality of cos (t) ", so the baud rate doubles)
In practice, because the actual bandwidth B is greater than the Nyquist bandwidth W, the bandwidth utilization of the actual FDM system is lower than the ideal situation.

"In this case, we can finally Tuqiongbijian the sub-carrier interval of OFDM to the Nyquist bandwidth, i.e. (without considering the next two sub-carriers),OFDM achieves the ideal channel frequency band utilization ."

Figure 32:OFDM Orthogonal sub-carrier, carrier spacing is nyquist bandwidth, ensuring maximum bandwidth utilization


The above-mentioned frequency domain analysis is equipped with the implementation of LTE, the following conditions:
Note: This paragraph describes the need for a basic knowledge of the LTE physical layer.

Sub-carrier interval δf=15khz, the transmission time of an OFDM symbol is 66.7US. On the 10MHz channel, the sub-frame of 1ms transmits 14 OFDM symbol "Not 15, left blank to CP", each OFDM symbol carries 600 plural information, therefore:
1. From the overall system, the baud rate is 600*14*2/1ms = 16.8MBaud, which occupies 10MHz bandwidth, so the bandwidth utilization is 16.8mbaud/10mhz = 1.68baud/hz, which is close to the 2baud/hz ideal situation. Note: One is that CP occupies about 1/15 of each OFDM symbol, and the second is that the 10MHz band is not uttered for transmitting data, and its boundary band needs to be left blank to reduce interference with neighboring channels. "
2. Single from the OFDM one symbol, the baud rate is 600*2/66.7us = 18MBaud, occupy the bandwidth 600*15khz=9mhz "not consider the boundary sub-carrier out-of-band problem", so its bandwidth utilization is 18mbaud/9mhz=2baud/hz, Meet the above discussion.
Attached: 5M bandwidth of WCDMA chip rate = 3.84m/s, that is, the bitrate is 3.84m*2 = 7.68MBaud, bandwidth 5M, so the bandwidth utilization is 7.68mbaud/5mhz = 1.536baud/hz, slightly inferior to the LTE 1.68baud/ Hz "NOTE: WCDMA pulse molding uses a roll-down coefficient of 0.22 of the L-cosine filter, Nyquist bandwidth of 3.84M"

Chapter Three: Realizing OFDM with Ifft

In fact, in the first two chapters, I have already expressed my understanding: The first section is the principle of sub-carrier orthogonal from the time domain, and the second section is to explain the characteristics of the optimal frequency band utilization after the sub-carrier Quadrature is explained from the frequency domain. Come to think of it, although the first two chapters have been written long "not expected to write so long ... It's too long for nobody to see ... but it should be simple, clear and understandable.
However, the "small white" jam, it does not seem to be the most basic orthogonal principle and frequency band utilization, but is the IFFT transformation, flooding in the various time domain frequency domain role transformation let its dazzling.

Personally feel to understand Ifft realize OFDM, the best way is to look at the formula. For example, in the first chapter of the equation 1-1 and equation 1-2, with the time-domain waveform overlay, push push to push a good understanding. Of course, the ifft here need to discretization the time domain, so the formula IFFT≈IDFT--

fn = 1/n ∑fk E (j 2π k n/n) "formula 3-1,n the sequence number after the field is discrete, N is the total number of IFFT, N∈[1,n]"

On the understanding of Equation 3-1, One way of understanding is to contact the first chapter of the formula 1-2: You can find the formula 3-1 equals to the right of the expression of the physical meaning and formula 1-2 is the same, all represent the different sub-carrier E (j 2π K n/n) send their respective signal FK, Then the superposition on the time domain forms FN, except that the time domain superimposed now is not a continuous waveform, but rather a discrete sampling point of timing.
　　Another way of understanding is : In an OFDM symbol time long T, with n subcarrier each send a signal f (k) (K∈[1,n]), equivalent to the direct in the time domain to send fn (n∈[1,n]) N signal, each signal sent t/n length.

In the implementation of OFDM in Ifft, the Ifft module is added to the transmitter, and the FFT module is added on the receiving side. the function of the Ifft module is equivalent to saying: Don't bother sending n subcarrier signals, I'll figure out what you're going to stack up in the air .the function of the FFT module is equivalent to saying: Don't use the old-fashioned integration method to remove the remaining orthogonal sub-carriers, Let me help you figure out all the N-carrying signals at once . In this way, Ifft realizes the OFDM system with " mathematical Method ", calculates the superposition waveform of the signal at the transmitting end, removes the orthogonal sub-carrier at the receiving end, thus greatly simplifies the system complexity.

Figure Eight: OFDM is implemented with IFFT. Compare your own figure seven


Finally say: "Small white" is "white rich beauty" of "white", Non-"poor white" of "white" also.
Well, it's over. It was longer and no one looked.

Supplemental Chapters: Orthogonality from the spectral perspective this article was first published without this paragraph, because the original text has been very self-consistent, the principle of OFDM has been very clear in place.　　However, this section of the content is in other articles on the OFDM often appear in the bridge section, so it is necessary to add a statement of their own views. Note: This section is a supplemental section, which is not necessarily related to the logic of this article and can be skipped directly. From the text section, you can find The author's idea: to explain the orthogonality of sub-carriers from the time domain angle, and to explain the frequency band utilization of OFDM。 The author finds this to be the easiest way to understand the principle of OFDM. But in the textbook and on the Internet, there is also a very mainstream way of explaining: "Intuitive" view of the orthogonality of sub-carriers from the frequency domain。 For example, the following diagram: Figure 51: Looking at orthogonality from the OFDM spectrum (this image is from the web, better than the picture I've drawn, and the textual explanation) this argument is: at each sub-carrier sampling point, the other sub-carrier signal sampling value is 0(That is, the subcarrier nulls corresponds to the subcarrier Peak of a sub-carrier). This statement has a very eye-catching visual effect on the illustrations, so it is a regular in the textbook handouts, but at least from the author's point of view, This argument becomes very difficult to understand and explain when it comes to the subsequent demodulation signal。　　So the first version of this article is not intended to write a subsection.　　If you see this, you feel just kind, so congratulations. 　　If you see this and think that it has made your head paste, then you can review the first chapter: Orthogonality on the time domain, and then continue reading the following section to detoxify. How to relate the relationship between orthogonality on time domain and orthogonality in frequency domain? Recalling that sin (t) and sin (2t) are orthogonal "proofs: sin (t) sin (2t) has an integral of 0" in the interval [0,2π], which is generalized to a more general case: {sin (2π δf t), sin (2π δf 2t), sin (2π δf 3t ,..., sin (2π δf kt)} is orthogonal on the interval [0, 1/δf] (note: The textbook is generally written as U (t) in the [-T/2,T/2] interval, this article does not need to be so academic). As can be seen, here is a key parameter δf: It is not only the space between the sub-carriers in the frequency domain, but also determines the time of signal transmission in the time domain. Review time domain frequency domain conversion diagram: Figure 52: The time-frequency conversion of the time-domain waveform and the frequency-domain conversion in the previous figure 21, can be found Δf Both determine the sub-carrier itself (that is, the first row of the two figures), but also determine the transmission time of the signal to be sent (that is, the width of the signal in the second row of two graphs), thus determining the main lobe width of the signal spectrum and the position of the Sidelobe 0。 This also means that once the sub-carrier interval is selected in the OFDM system, the orthogonality in the time domain and the orthogonality in the frequency domain are logically connected. such as: Figure 53: The same as the previous figure 23, the two-way signal interval δf, to ensure the orthogonal in the time domain, to determine the frequency domain of the sidelobe 0 point position in fact, the author, from the spectrum to see the orthogonality of OFDM a little reversal of causal suspicion. As I understand it: OFDM chooses the orthogonal sub-carrier is because, the spectrum appears "the remainingSubcarrier carrying SignalThe Sidelobe 0 point is in the currentSubcarrier carrying SignalAt the peak of the main lobe. phenomenais the fruit. It is a fallacy to push the cause of the fruit.
Reference [1]: Wireless Communications, Andrea Goldsmith-12.2 multicarrier modulation with overlapping subchannels

Reference [2]: Principles of Digital communication-gallager-6.4.1 Double-sideband amplitude modulation

OFDM Understanding (reprint)