### 滤波法与频谱法在瑞利衰落信道仿真中的比较研究

2021-08-06俞骋

俞骋

（宁波第二技师学院，宁波315012）

## 1 Introduction

The aim of this article is to study and get a good understanding of wireless channel and the factors which affects the performance of such a channel.This is done by simulating a Rayleigh fading channel with Clarke’s Doppler spectrum.Simulations are done in MATLAB.Two differ⁃ent methods are used to analyze the performance of the fading channel:the filter method and the spectrum method.

## 2 The Filter Method

In the filter method,c（t）is considered as the output from a filter with impulse response g（t）when the incom⁃ing signal x（t）is taken as complex,white Gaussian noise with unit variance.The output of the filter will simply be the convolution of x（t）and g（t）:

The impulse response function of the filter is chosen as

where fDis the Doppler frequency calculated as

where v is the relative velocity between the transmit⁃ter and the receiver,fcthe carrier frequency and c0the speed of light.To compute ĝ(t)we have chosen a timelimited window function w（n）whose value is being set equal to 1 in a finite interval,so that ĝ(t)can be made causal and time limited.Since Matlab is being used,dis⁃crete time samples are needed.Therefore,impulse re⁃sponse ĝ(nTs)is calculated as

The factor K was chosen（was tuned such that）such that ĝ(nTs)has unit power.Figure 1 shows ĝ(nTs)for a signal with carrier frequency 2 GHz and a relative speed of 50 km/h,which gives us a Doppler frequency fD=92 Hz.

The input signal z（t）to the filter was generated with zero mean,unit variance,complex Gaussian distribution.The impulse response of the channel was finally calculat⁃ed as:

The magnitude,phase and power spectral density of the impulse response of the channel is shown in figure 1 and 2 respectively.The power spectral density of the chan⁃nel was calculated by using the autocorrelation ofc（t）.

Figure 1 Amplitude and phase of c（t）

Figure 2 Power Spectral Density of c（t）

## 3 The Spectrum Method

In the spectrum method the fading gain is sampled in the frequency domain instead of the time domain.The sam⁃ples are then transformed to the time domain using an in⁃verse FFT.

First a Clarke's Doppler spectrum was constructed with power spectral densityS c(f)according to:

From this spectrumG(f)was easily calculated asThe function was then periodically shifted toG͂(f)=G(f)+G(f-f s)and sampled withN=10000 samples fromftof s.This shift is illustrated in figures 4 and 5.

The next step was to draw N=10000 statistically inde⁃pendent,zero-mean,complex Gaussian random numbers with a variance that results in a unit variance impulse re⁃sponsec（t）at the end.The implementation of this vari⁃ance was built on a trial and error approach.

Finally,c（t）was calculated according to:

whereZ(k)is the Gaussian noise.The resulting am⁃plitude and phase of the impulse response is showed in fig⁃ure 3.The power spectral density of the channel was calcu⁃lated by using the autocorrelation ofc（t）,and it is showed in figure 4.

Figure 3 Amplitude and phase of c（t）

Figure 4 Power spectral density of c（t）

## 4 Time and frequency-varying Channel

A study was also made on a Rayleigh fading channel with both time and frequency variations.The model weused for simulating this channel is built onL=3 taps with complex gainc l(t)where the received signal is calculated according to:

The responseC（f，n T s）for such a channel is just the amplitude scaling a complex exponential with frequency f experiences when it is transmitted over the channel.To compute the time-varying frequency response we used the equation.

wherec l(nT s)was calculated using the filter method.A plot of the response was simulated for each and one of the nine combinations forL=1，2，3 andf D T s=0.0005,0.005,0.05.The behaviour of the time and frequency re⁃sponse of the channel for a specific case can be observed in figure 5.

Figure 5 Time and frequency varying channel response for L=3 and f D T s=0.005

## 5 Discussion

### 5.1 Chan nel characteristics for the filter-and spectrum method

After simulating channel responses with the two meth⁃ods we compared the amplitude and phase of c（t）with the⁃ory to roughly verify that they are valid.The characteris⁃tics of these plots matched the theory quite well.A more detailed analysis was made based on autocorrelation,PSD,level crossing rate,average fade duration,complexity etc.

（1）Autocorrelation and power spectral density

The PSD of the channel impulse responseS c(f)with the filter and spectrum method was plotted by taking Fouri⁃er transform of its autocorrelation function.The results ob⁃tained from the two methods are quite similar.By compar⁃ing the PSD and autocorrelation plots of the channel ob⁃tained from the two methods we couldn’t say everything about the channel characteristics.Therefore,level cross⁃ing rate and average fade duration was calculated through simulation and theoretically as well.

（2）Level crossing rate

Another parameter we used to verify the characteris⁃tics of the channel was the level crossing rate.Level cross⁃ing rate was calculated by setting different threshold lev⁃els.The values are obtained for 20,000 samples and run⁃ning the program for 5 times to average it out.Table 1 re⁃veals that both methods generate a channel with quite simi⁃lar response to what theory predicts.

Table 1 Level Crossing Rate

Table 2 Average Fade Duration

（3）Average fade duration

As a complement to the level crossing rate the average fade duration was calculated for both channels,see table 2.Also here the two methods generate similar results.

（4）Advantages and disadvantages

As one can note above from the parameters,both methods generates a channel response close to the theory.The filter method is however a bit more complex and it takes more time to simulate a channel.This was tested by increasing the number of samples while noting the time for simulation.But it gives better results compared to spec⁃trum method when number of samples is taken very large.

By increasing the length of the input signal Ns the re⁃sult would definitely be better since the input is a random process and we are going to estimate the channel which is arandom process itself.In other words:the more the Ns is,the better the channel estimation would be.But the draw⁃back of this is that by increasing Ns the simulation will take longer time.The other important factor is the length of the filter N which is a Bessel function.By getting more sam⁃ples of the Bessel function with truncation method the filter would be better estimated so the result would be better.

### 5.2 Time and frequency-varying channel

If we have only one tap,i.e.L=1,our channel will have less delay spread,but on the other hand the channel performance will suffer.By increasing the number of taps（L=2,L=3）delay spread of the channel will increase but at the same time channel performance will get better.There is always a tradeoff between channel performance and complexity.

The factor fDTsshows how fastly a channel is vary⁃ing in time.At low value of fDTsthe channel will vary very slowly,where as for higher values of fDTschannel variations will increase as can be seen from the plots ap⁃pended in Appendix B.

When we have one tap L=1 and normalized Doppler frequency fDTs=0.0005 our channel will behave like slow,flat fading channel which is quite obvious from the plot in Appendix B.By increasing the Doppler frequency to fDTs=0.005,0.05 channel variations will also increase.