Multi-Path Fading detailed

Conclusion: Bandwidth is less than coherent bandwidth, flat fading, and channel selective fading.

The symbol period is less than coherence time, slow fading, and vice versa fast fading.

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one of the simplest programs for you to understand Multipath channels

Time-varying, multipath is the characteristics of wireless channels, I believe many people after reading a lot of books, the wireless channel feeling or confused. Why Multipath leads to frequency selectivity. Why the Doppler frequency shift reflects the time-varying of the channel. There is certainly a sense of confusion about these problems. Let's use a simple no-no-go program one by one to solve your puzzle.

First, let's first talk about the scenario of the program simulation. As shown in Figure 1.

Figure 1 The simplest multipath channel

Suppose a fixed base station is installed at one end of a straight freeway, and at the other end there is a wall that completely reflects the electromagnetic wave, and the distance from the base station to the reflective wall is d. The initial distance from the base station to the mobile stations is r0. The base station emits a sinusoidal signal with a frequency of F, expressed as a cos (2*pi*f*t). Due to the reflection of the wall, the mobile station can receive a 2-path signal, one of which is a direct transmission from the base station signal, the other is reflected from the reflective wall signal.

OK, first of all we look at the static situation of the mobile station. It is clear that the time required to reach the mobile station from the direct signal from the BTS is R0/C (c is the speed of light), and the time required to reach the mobile station from a signal reflected from the reflective wall is (d+d-r0)/c= (2D-R0)/C. In other words, at the moment T, the Mobile station receives a direct signal from the time (T-R0/C) base station and a reflection signal from the time t (2D-R0)/C base station, respectively. We know that the signal in the process of transmission to decay, in free space, the electromagnetic wave power with the distance r by the square Law attenuation, the corresponding electric field intensity (can be regarded as the receiving signal voltage) with the 1/R law attenuation. And the reflected signal is opposite the phase of the direct signal. So, at the moment the T-Mobile station receives the synthesized signal as

E (t) =

The minus sign reflects the opposite phase of the reflected signal to the direct signal.

What is the characteristic of receiving signals at R0. Let's draw it out. Here is the program code.

Clear All

f=1; % transmit signal frequency

v=0; % Mobile station speed, quiescent condition is 0

C=3e8; % electromagnetic velocity, speed of light

r0=3; % Mobile Station initial distance from base stations

d=10; % base station distance from reflective wall

T1=0.1:0.0001:10; % Time

E1=cos (2*pi*f* ((1-v/c) *t1-r0/c))./(R0+V.*T1);

E2=cos (2*pi*f* (1+v/c) *t1+ (r0-2*d)/C)./(2*D-R0-V*T1);

Figure

Plot (T1,E1)% draw a signal with a direct diameter

On

Plot (T1,-e2, '-G ')% to draw a signal of the reflection diameter

On

Plot (T1,e1-e2, '-R ')% draws the total receiving signal from the mobile station.

Legend (' Direct path signal ', ' reflection diameter signal ', ' mobile station ' received synthesized signal ')

Axis ([0 10-0.8 0.8])

After running the program, the results are as shown in Figure 2:

Fig. 2 Direct signal, reflection signal and synthesized signal when r0=3

wherein, the Blue line is the direct diameter signal, the green line is the reflection diameter signal, the red line is the mobile station received the 1th and 2nd diameter synthesis signal, from the figure we can see, even if the mobile station is still, due to the presence of the reflection diameter, so that the received synthesis signal maximum value is less than the direct diameter signal.

Now let's change the position of the mobile station from the base station, so that the r0=9, so that it closer to the location of the reflective wall, run the program again, the result is as shown in Figure 3:

Fig. 3 Direct signal, reflection signal and synthesized signal when r0=9

As can be seen from Figure 3, this time, due to the location of the reflective wall, the direct signal is weaker than the r0=3, the reflection signal is stronger than the signal at the r0=3 position, but the mobile station received the synthesis signal is weaker. Not only less than the direct diameter of the signal is less than the reflection diameter of the signal.

We can conclude that, even if the mobile station is still, due to the presence of the reflection diameter, the reception signal is weaker than the absence of the reflection diameter of the signal, the decline resulting.

Wait, we just fired a single 1-frequency signal in the experiment, and what happens when the other frequencies are emitted?

We changed the F=100e6 and r0=3 the result as shown in Figure 4. At this point, because F is too large, all points are connected in a straight line. But we can see that the red line is significantly larger than the blue and green lines, which means that the signal received by the mobile station is enhanced when f=100e6.

Fig. 4 Direct signal, reflection signal and synthesized signal when f=100e6

To make it more intuitive, let's change the speed of light C to c=10, so the change doesn't affect the substance of the discussion, but it can help us see it more intuitively. Let F=1 and f=5, respectively, draw the graph as shown in Figure 5, Figure 6:

Fig. 5 Direct signal, reflection signal and synthesized signal when f=1,c=10

Fig. 6 Direct signal, reflection signal and synthesized signal at f=5, c=10

From the results we can clearly see that this time the f=1 signal is enhanced, and the f=5 signal is weakened.

I think, here, you should understand how frequency-selective fading is going on. In the same position, due to the presence of a reflection path signal, the frequency at which the signal is received at the receiver is enhanced, and some frequencies are weakened. Frequency selectivity arises from this.

Since there is frequency-selective fading, we naturally ask what frequencies will be enhanced and which frequencies will be weakened.

In the example above, if we let c=3e8, let f=1,2,3 ... ,..., 1000, you will find that these frequencies are basically weakened, only if f is sufficiently large, such as f=100e6, to see that the signal is enhanced, then we call the frequency range that is basically consistent with the influence of the coherent bandwidth. How coherent bandwidth is obtained.

As can be seen from our experiments, the receiving signal is the superposition of 2 radio waves with a frequency of F, the phase difference between the 2 waves is:

From this formula, we can see that for a fixed r, if f changes, then the synthesized signal from the peak to reach the trough, and exactly is the reflection diameter and the direct path of the propagation delay difference. If the frequency changes far less than 1/td, then the signal is enhanced or weakened without noticeable changes. Therefore, the parameter 1/td is called the coherent bandwidth.

With the concept of coherent bandwidth, we look at flat fading and frequency selective fading.

Assuming that the transmitted signal bandwidth is narrower and less than the coherent bandwidth, we can know that the fading effect in the signal frequency band is basically the same. At this time, this decline is called flat decline. All know that the frequency band is narrower, which means that the time domain signal pulse period is longer, when the signal bandwidth is exactly equal to the coherent bandwidth, it can be approximated that the signal pulse period is approximately equal to the difference of propagation delay. At this point, when the mobile station just receives the 2nd pulse of the direct diameter, the 1th pulse from the reflection path arrives at the same time, so the synthesized signal is the 2nd pulse of the direct diameter and the 1th pulse of the reflection diameter. See here, we will understand how the inter-code interference is produced. If we increase the signal pulse period, the corresponding signal band narrows, then the inter-code interference will become smaller. It is said that the 1th pulse of the reflection path arrives, the 1th pulse of the direct path is not finished. The longer the pulse period, the greater the number of pulses coincident with the direct and reflective diameters, the lighter the interference between the codes. When the pulse period is Yushijencha, we can approximate the signal of the direct diameter and the signal of the reflection diameter as the same diameter signal. Of course, the pulse amplitude of the signal will change. Rayleigh fading arises when we think of the more reflective path as the same path signal. In the case of more reflective diameters, the direction of each path is different and the phase is different, which can be regarded as a random variable subjected to the same distribution. By the knowledge of probability theory, multiple obey the same distributed random variable and obey the Gaussian distribution. As the actual signal is generally transmitted through the I, Q Two, so I-way obeys the Gaussian distribution, Q Road obeys the Gaussian distribution, the envelope obeys the Rayleigh distribution. Look at the definition of the Rayleigh distribution.

Above is to discuss the signal pulse period is larger than the propagation delay, and then discuss the signal pulse period is less than the propagation delay situation. According to the time-frequency relationship we can know that the pulse period is short, which means that the signal band becomes wider than the coherent bandwidth. It has been said that the effect of frequency is different when the bandwidth is greater than the coherence. So the decline of this time is frequency selective fading. Considering the time domain, the pulse period becomes shorter. Assuming a 1/2 propagation delay, the 1th pulse of the reflection path arrives when the mobile station receives the 3rd pulse of the direct path. It is obvious that the 1th pulse of the reflection diameter interferes with the 3rd pulse of the direct path. It is not considered that the signal of the direct diameter and the reflection diameter is the same diameter signal. When the pulse period is shortened further, the frequency selective fading is more serious when the corresponding signal frequency band is further enlarged. It is conceivable that, in the presence of more reflective path, inter-code interference will be more serious.

Here, you should understand the relationship between multipath channels and Rayleigh fading and frequency selective fading. Now let's look at the time-variant of the channel.

This paper discusses the static situation of the mobile station. Now let's move the mobile station to the reflective wall at a speed of V. Then at the moment T, the Mobile station is r=r0+v*t from the base station position. Replace the r0 in the 1th formula with R:

Let's first draw the signal received over time. The program code is the same as above, but in order to make the time denaturation more intuitive, we let c=10, while V is no longer 0,v=1,f=2,d=15,t1=0.1:0.001:12; After running the program, you will see the following figure (Figure 7):

Fig. 7 Direct signal, reflected signal and synthesized signal when moving station movement

We separately draw the receiving signal separately. Use the command:

Figure;plot (T2,e1-e2, '-R ')

The result is shown in Figure 8.

Fig. 8 Synthesis signal when moving station movement

In the previous procedure, we saw that multipath led to frequency selectivity. When the mobile station is moving, we find that even the same frequency, at different points in time, the strength of the synthesized signal is not the same. In Figure 8, we can see that at t=2,4.5,7,9.5s, the intensity of the received signal is relative to the trough position, especially when t=9.5s, the received synthesis signal is almost 0, and we control the t=9.5s when the direct signal and reflection signal, they are much larger than the synthetic signal. In the t=3,5.5,8,10.5 position, the intensity of the received signal is relative to the crest position. The situation of the signal enhancement or weakening caused by the movement of the mobile station is the time selective fading. Why does exercise produce time-selective fading?

Let's look at the 2nd formula. The 1th direct wave is a sine wave with a frequency of f (1-v/c), which undergoes a doppler shift of:; the 2nd is a sine wave with a frequency of f (1+v/c), and the Doppler shift experienced is:, parameter

Called the Doppler extension. For example in the above program, f=2,v=1,c=10, so ds=0.4. The effect of Doppler expansion is most easily observed when the distance between the mobile station and the reflective wall is closer than the distance from the transmitting antenna. In this case, the attenuation of the 2 paths is approximately the same, so that the denominator of the 2nd item in the R=R0+VT approximation formula can be used, thus combining 2 sine signals to obtain:

This is the product of 2 sinusoidal signals, where 1 signals have an input frequency of f, usually a GHz order of magnitude, and another signal frequency is FV/C=DS/2, so the response to a sinusoidal signal with a frequency of f is another sinusoidal signal of F, which has a time-varying envelope, Every 2.5s from the trough to the crest and then to the trough. When the mobile station is in the crest position, the received signal is enhanced, while at the trough position, the signal is attenuated. Now you can see why the denominator entries in the 2nd formula can be partially ignored. When the difference of 2 path length changes 1/4 wavelengths, the phase difference of the response signal of these 2 paths changes pi/2, resulting in a very serious change in the total reception amplitude. Due to the fact that the length of the carrier wave is very small for the path, the time when the amplitude changes significantly by this phase effect is much less than that caused by the denominator term. In Fig. 7, for example, we can see that the amplitude of the direct signal and the reflected signal has not changed greatly during the period of t=9.5 to t=12, but the amplitude of the synthesized signal received is greatly fluctuating due to the change of the direct and reflected signal phase. So during this period of t=9.5 to t=12, we can assume that the denominator in equation 2 is constant.

Let's take a look at Figure 8, although from t=9.5 to t=12, the received signal amplitude has undergone a shift from trough to crest to trough, but if we observe t=10.5 to t=11.5 this time, we can find that the signal amplitude is basically the same. We refer to the time period in which the channel is essentially unchanged as the coherent time () of the channel. Here, I think you should understand why the coherence time of the channel is related to the Doppler shift, the greater the Doppler, the shorter the coherent time of the channel. We can let F=4 run the program again, and the result is shown in Figure 9. This time you will find that the synthetic signal envelope changes faster, and the channel coherence time correspondingly becomes smaller. The coherence time simply says that the characteristics of the channel during this period are basically unchanged. Whether the signal is enhanced or weakened during this time is not reflected.

Fig. 9 Synthesis signal at f=4

We know that in digital communication, the receiving end is periodic to the receiving symbol to restore information, the period of 1 symbol pulses can be very small, so according to the correlation time and the symbol pulse period relative length, we can divide the channel into slow-change channel and fast-change channel. For example, in Figure 8, if the period of sending symbols is less than 1.25s, we can think that this is a slow-change channel (or quasi-static channel) if the sending symbol period is greater than 1.25s, in the process of sending symbols, the channel characteristics have changed significantly, we think this is a fast-change channel. So the fast change of the channel or the slow change is relative to the period of the sending symbol.

At this point, we discuss the time-variant of the channel, combined with the frequency selectivity discussed earlier, the wireless channel can be divided into 4 kinds: slow-change Rayleigh fading channel, fast-change Rayleigh fading channel, slow frequency selective channel, fast variable frequency selective channel.

Well, our class is over, I hope you can read it to solve your doubts. Goodluck. ^_^.