A brief intro to modeling a frequency selective fading channel using tapped delay line (TDL) filters. Rayleigh & Rician frequency-selective fading channel models explained.
Tapped delay line filters
Tapped-delay line filters (FIR filters) are best to simulate multiple echoes originating from same source. Hence they can be used to model multipath scenarios. Tapped-Delay-Line (TDL) filters with number taps can be used to simulate a multipath frequency selective fading channel. Frequency selective channels are characterized by time varying nature of the channel. For simulating a frequency selective channel, it is mandatory to have N > 1. In contrast, if N = 1, it simulates a zero-mean fading channel where all the multipath signals arrive at the receiver at the same time.
Let be the associated path attenuation corresponding to the received power and propagation delay of the th path. In continuous time, the complex path attenuation is given by
The complex channel response is given by
In the equation above, the attenuation and path delay vary with time. This simulates a time-variant complex channel.
As a special case, in the absence any movements or other changes in the transmission channel, the channel can remain fairly time invariant (fixed channel with respect to instantaneous time ) even though the multipath is present. Thus the time-invariant complex channel becomes
Usually, the pair is described as a Power Delay Profile (PDP) plot. A sample power delay profile plot for a fixed, discrete, three ray model with its corresponding implementation using a tapped-delay line filter is shown in the following figure
Figure 1: 3-ray multipath time-invariant channel and its equivalent TDL implementation (path attenuations and propagation delays are fixed)
Choose underlying distribution:
The next level of modeling involves, introduction of randomness in the above mentioned model there by rendering the channel response time variant. If the path attenuations are typically drawn from a complex Gaussian random variable, then at any given time , the absolute value of the impulse response is
● Rayleigh distributed – if the mean of the distribution ● Rician distributed – if the mean of the distribution
Respectively, these two scenarios model the presence or absence of a Line of Sight (LOS) path between the transmitter and the receiver. The first propagation delay has no effect on the model behavior and hence it can be removed.
Similarly, the propagation delays can also be randomized, resulting in a more realistic but extremely complex model to implement. Furthermore, the power-delay-profile specifications with arbitrary time delays, warrant non-uniformly spaced tapped-delay-line filters, that are not suitable for practical simulation. For ease of implementation, the given PDP model with arbitrary time delays can be converted to tractable uniformly spaced statistical model by a combination of interpolation/approximation/uniform-sampling of the given power-delay-profile.
Real-life modelling:
Usually continuous domain equations for modeling multipath are specified in standards like COST-207 model in GSM specification. Such continuous time power-delay-profile models can be simulated using discrete-time Tapped Delay Line (TDL) filter with number of taps with variable tap gains. Given the order , the problem boils down to determining the discrete tap spacing and the path gains , in such a way that the simulated channel closely follows the specified multipath PDP characteristics. A survey of method to find a solution for this problem can be found in [2].
Rate this article: Note: There is a rating embedded within this post, please visit this post to rate it.
Small-scale Models for Multipath Effects
● Introduction
● Statistical characteristics of multipath channels
□ Mutipath channel models
□ Scattering function
□ Power delay profile
□ Doppler power spectrum
□ Classification of small-scale fading
● Rayleigh and Rice processes
□ Probability density function of amplitude
□ Probability density function of frequency
● Modeling frequency flat channel
● Modeling frequency selective channel
□ Method of equal distances (MED) to model specified power delay profiles
□ Simulating a frequency selective channel using TDL model
Power delay profile gives the signal power received on each multipath as a function of the propagation delays of the respective multipaths.
Power delay profile (PDP)
A multipath channel can be characterized in multiple ways for deterministic modeling and power delay profile (PDP) is one such measure. In a typical PDP plot, the signal power on each multipath is plotted against their respective propagation delays.
In a typical PDP plot, the signal power () of each multipath is plotted against their respective propagation delays (). A sample power delay profile plot, shown in Figure 1, indicates how a transmitted pulse gets received at the receiver with different signal strengths as it travels through a multipath channel with different propagation delays. PDP is usually supplied as a table of values, obtained from empirical data and it serves as a guidance to system design. Nevertheless, it is not an accurate representation of the real environment in which the mobile is destined to operate at.
Figure 1: A typical discrete power delay profile plot for a multipath channel with 3 paths
The PDP, when expressed as an intensity function , gives the signal intensity received over a multipath channel as a function of propagation delays. The PDP plots, like the one shown in Figure 1, can be obtained as the spatial average of the complex channel impulse response as
RMS delay spread and mean delay
The RMS delay spread and mean delay are two most important parameters that characterize a frequency selective channel. They are derived from power delay profile. The delay spread of a multipath channel at any time instant, is a measure of duration of time over which most of the symbol energy from the transmitter arrives the receiver.
In the wide-sense stationary uncorrelated scattering (WSSUS) channel models [1], the delays of received waves arriving at a receive antenna are treated as uncorrelated. Therefore, for the WSSUS model, the underlying complex process is assumed as zero-mean Gaussian random proces and hence the RMS value calculated from the normalized PDP corresponds to standard deviation of PDP distribution.
Figure 2: Relation between scattering function, power delay profile, Doppler power spectrum, spaced frequency correlation function and spaced time correlation function
For continuous PDP (as in Figure 2), the RMS delay spread () can be calculated as
where, the mean delay is given by
For discrete PDP (as in Figure 1), the RMS delay spread () can be calculated as
where, is the power of the path, is the delay of the path and the mean delay is given by
Knowledge of the delay spread is essential in system design for determining the trade-off between the symbol rate of the system and the complexity of the equalizers at the receiver. The ratio of RMS delay spread () and symbol time duration () quantifies the strength of intersymbol interference (ISI). Typically, when the symbol time period is greater than 10 times the RMS delay spread, no ISI equalizer is needed in the receiver. The RMS delay spread obtained from the PDP must be compared with the symbol duration to arrive at this conclusion.
Frequency selective and non-selective channels
With the power delay profile, one can classify a multipath channel into frequency selective or frequency non-selective category. The derived parameter, namely, the maximum excess delay together with the symbol time of each transmitted symbol, can be used to classify the channel into frequency selective or non-selective channel.
PDP can be used to estimate the average power of a multipath channel, measured from the first signal that strikes the receiver to the last signal whose power level is above certain threshold. This threshold is chosen based on receiver design specification and is dependent on receiver sensitivity and noise floor at the receiver.
Maximum excess delay, also called maximum delay spread, denoted as (), is the relative time difference between the first signal component arriving at the receiver to the last component whose power level is above some threshold. Maximum delay spread () and the symbol time period () can be used to classify a channel into frequency selective or non-selective category. This classification can also be done using coherence bandwidth (a derived parameter from spaced frequency correlation function which in turn is the frequency domain representation of power delay profile).
Maximum excess delay is also an important parameter in mobile positioning algorithm. The accuracy of such algorithm depends on how well the maximum excess delay parameter conforms with measurement results from actual environment. When a mobile channel is modeled as a FIR filter (tapped delay line implementation), as in CODIT channel model [2], the number of taps of the FIR filter is determined by the product of maximum excess delay and the system sampling rate. The cyclic prefix in a OFDM system is typically determined by the maximum excess delay or by the RMS delay spread of that environment [3].
A channel is classified as frequency selective, if the maximum excess delay is greater than the symbol time period, i.e, . This introduces intersymbol interference into the signal that is being transmitted, thereby distorting it. This occurs since the signal components (whose powers are above either a threshold or the maximum excess delay), due to multipath, extend beyond the symbol time. Intersymbol interference can be mitigated at the receiver by an equalizer.
In a frequency selective channel, the channel output can be expressed as the convolution of input signal and the channel impulse response , plus some noise .
On the other hand, if the maximum excess delay is less than the symbol time period, i.e, , the channel is classified as frequency non-selective or zero-meanchannel. Here, all the scattered signal components (whose powers are above either a specified threshold or the maximum excess delay) due to the multipath, arrive at the receiver within the symbol time. This will not introduce any ISI, but the received signal is distorted due to inherent channel effects like SNR condition. Equalizers in the receiver are not needed. A time varying non-frequency selective channel is obtained by assuming the impulse response . Thus the output of the channel can be expressed as
Therefore, for a frequency non-selective channel, the channel output can be expressed simply as product of time varying channel response and the input signal. If the channel impulse response is a deterministic constant, i.e, time invariant, then the non-frequency selective channel is expressed as follows by assuming
This is the simplest situation that can occur. In addition to that, if the noise in the above equation is white Gaussian noise, the channel is called additive white Gaussian noise (AWGN) channel.
Rate this article: Note: There is a rating embedded within this post, please visit this post to rate it.
Small-scale Models for Multipath Effects
● Introduction
● Statistical characteristics of multipath channels
□ Mutipath channel models
□ Scattering function
□ Power delay profile
□ Doppler power spectrum
□ Classification of small-scale fading
● Rayleigh and Rice processes
□ Probability density function of amplitude
□ Probability density function of frequency
● Modeling frequency flat channel
● Modeling frequency selective channel
□ Method of equal distances (MED) to model specified power delay profiles
□ Simulating a frequency selective channel using TDL model
Understand various characteristics of a wireless channel through multipath channel models. Discuss Wide Sense Stationary channel, uncorrelated scattering channel, wide sense stationary uncorrelated scattering channel models and scattering function.
Introduction
Wireless channel is of time-varying nature in which the parameters randomly change with respect to time. Wireless channel is very harsh when compared to AWGN channel model which is often considered for simulation and modeling. Understanding the various characteristics of a wireless channel and understanding their physical significance is of paramount importance. In these series of articles, I seek to expound various statistical characteristics of a multipath wireless channel by giving more importance to the concept than the mathematical derivation.
In a multipath channel, multiple copies of a signal travel different paths with different propagation delays and are received at the receiver at different phase angles and strengths. These rays add constructively or destructively at the receiver front end, thereby giving rise to rapid fluctuations in the channel. The multipath channel can be viewed as a linear time variant system where the parameters change randomly with respect to time. The channel impulse response is a two dimensional random variable – that is a function of two parameters – instantaneous time and the propagation delay . The channel is expressed as a set of random complex gains at a given time and the propagation delay . The output of the channel can be expressed as the convolution of the complex channel impulse response and the input
If the complex channel gains are typically drawn from a complex Gaussian distribution, then at any given time , the absolute value of the impulse response is Rayleigh distributed (if the mean of the distribution or Rician distributed (if the mean of the distribution . These two scenarios model the presence or absence of a Line of Sight (LOS) path between the transmitter and the receiver.
Here, the values for the channel impulse response are samples of a random process that is defined with respect to time and the multipath delay τ. That is, for each combination of and τ, a randomly drawn value is assigned for the channel impulse response. As with any other random process, we can calculate the general autocorrelation function as
Given the generic autocorrelation function above, following assumptions can be made to restrict the channel model to the following specific set of categories
Wide Sense Stationary channel model
Uncorrelated Scattering channel model
Wide Sense Stationary Uncorrelated Scattering channel model
Wide Sense Stationary (WSS) channel model
In this channel model, the impulse response of the channel is considered Wide Sense Stationary (WSS) , that is the channel impulse response is independent of time . In other words, the autocorrelation function is independent of time instant \(t\) and it depends on the difference between the time instants where and . The autocorrelation function for WSS channel model is expressed as
Uncorrelated Scattering (US) channel model
Here, the individual scattered components arriving at the receiver front end (at different propagation delays) are assumed to be uncorrelated. Thus the autocorrelation function can be expressed as
Wide Sense Stationary Uncorrelated Scattering (WSSUS) channel model
The WSSUS channel model combines the aspects of WSS and US channel model that are discussed above. Here, the channel is considered as Wide Sense Stationary and the scattering components arriving at the receiver are assumed to be uncorrelated. Combining both the worlds, the autocorrelation function is
Scattering function
The autocorrelation function of the WSSUS channel model can be represented in frequency domain by taking Fourier transform with respect one or both variables – difference in time and the propagation delay . Of the two forms, the Fourier transform on the variable gives specific insight to channel properties in terms of propagation delay and the Doppler Frequency simultaneously. The Fourier transform of the above two-dimensional autocorrelation function on the variable is called scattering function and is given by
Fourier transform of relative time is Doppler Frequency. Thus, the scattering function is a function of two variables – Dopper Frequency and the propagation delay . It gives the average output power of the channel as a function of Doppler Frequency and the propagation delay .
Two important relationships can be derived from the scattering function – Power Delay Profile (PDP) and Doppler Power Spectrum. Both of them affect the performance of a given wireless channel. Power Delay Profile is a function of propagation delay and the Doppler Power Spectrum is a function of Doppler Frequency.
Power Delay Profile gives the signal intensity received over a multipath channel as a function of propagation delays. It is obtained as the spatial average of the complex baseband channel impulse response as
Power Delay Profile can also be obtained from scattering function, by integrating it over the entire frequency range (removing the dependence on Doppler frequency).
Similarly, the Doppler Power Spectrum can be obtained by integrating the scattering function over the entire range of propagation delays.
Fourier Transform of Power Delay Profile and Inverse Fourier Transform of Doppler Power Spectrum:
Power Delay Profile is a function of time which can be transformed to frequency domain by taking Fourier Transform. Fourier Transform of Power Delay Profile is called spaced-frequency correlation function. Spaced-frequency correlation function describes the spreading of a signal in frequency domain. This gives rise to the importance channel parameter – Coherence Bandwidth.
Similarly, the Doppler Power Spectrum describes the output power of the multipath channel in frequency domain. The Doppler Power Spectrum when translated to time-domain by means of inverse Fourier transform is called spaced-time correlation function. Spaced-time correlation function describes the correlation between different scattered signals received at two different times as a function of the difference in the received time. It gives rise to the concept of Coherence Time.
This website uses cookies to improve your experience while you navigate through the website. Out of these, the cookies that are categorized as necessary are stored on your browser as they are essential for the working of basic functionalities of the website. We also use third-party cookies that help us analyze and understand how you use this website. These cookies will be stored in your browser only with your consent. You also have the option to opt-out of these cookies. But opting out of some of these cookies may affect your browsing experience.
Necessary cookies are absolutely essential for the website to function properly. These cookies ensure basic functionalities and security features of the website, anonymously.
Cookie
Duration
Description
cookielawinfo-checbox-analytics
11 months
This cookie is set by GDPR Cookie Consent plugin. The cookie is used to store the user consent for the cookies in the category "Analytics".
cookielawinfo-checbox-analytics
11 months
This cookie is set by GDPR Cookie Consent plugin. The cookie is used to store the user consent for the cookies in the category "Analytics".
cookielawinfo-checbox-functional
11 months
The cookie is set by GDPR cookie consent to record the user consent for the cookies in the category "Functional".
cookielawinfo-checbox-functional
11 months
The cookie is set by GDPR cookie consent to record the user consent for the cookies in the category "Functional".
cookielawinfo-checbox-others
11 months
This cookie is set by GDPR Cookie Consent plugin. The cookie is used to store the user consent for the cookies in the category "Other.
cookielawinfo-checbox-others
11 months
This cookie is set by GDPR Cookie Consent plugin. The cookie is used to store the user consent for the cookies in the category "Other.
cookielawinfo-checkbox-necessary
11 months
This cookie is set by GDPR Cookie Consent plugin. The cookies is used to store the user consent for the cookies in the category "Necessary".
cookielawinfo-checkbox-performance
11 months
This cookie is set by GDPR Cookie Consent plugin. The cookie is used to store the user consent for the cookies in the category "Performance".
viewed_cookie_policy
11 months
The cookie is set by the GDPR Cookie Consent plugin and is used to store whether or not user has consented to the use of cookies. It does not store any personal data.
Functional cookies help to perform certain functionalities like sharing the content of the website on social media platforms, collect feedbacks, and other third-party features.
Performance cookies are used to understand and analyze the key performance indexes of the website which helps in delivering a better user experience for the visitors.
Analytical cookies are used to understand how visitors interact with the website. These cookies help provide information on metrics the number of visitors, bounce rate, traffic source, etc.
![\displaystyle{\tilde{a_n}(t) = a_n(t) exp\left[ -j 2 \pi f_c \tau_n(t) \right]}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle%7B%5Ctilde%7Ba_n%7D%28t%29+%3D+a_n%28t%29+exp%5Cleft%5B+-j+2+%5Cpi+f_c+%5Ctau_n%28t%29+%5Cright%5D%7D+&bg=ffffff&fg=000&s=2&c=20201002)
![E [h(t; \tau)] = 0](https://s0.wp.com/latex.php?latex=E+%5Bh%28t%3B+%5Ctau%29%5D+%3D+0&bg=ffffff&fg=000&s=0&c=20201002)
![E [h(t; \tau)] \neq 0](https://s0.wp.com/latex.php?latex=E+%5Bh%28t%3B+%5Ctau%29%5D+%5Cneq+0&bg=ffffff&fg=000&s=0&c=20201002)
![E\left [ h(t,\tau) \right ) \neq 0]](https://s0.wp.com/latex.php?latex=E%5Cleft+%5B+h%28t%2C%5Ctau%29+%5Cright+%29+%5Cneq+0%5D+&bg=ffffff&fg=000&s=0&c=20201002)
![R_{hh}(t_1,t_2;\tau_1,\tau_2)= E\left [ h(t_1,\tau_1)h^*(t_2,\tau_2) \right ] \quad\quad (2)](https://s0.wp.com/latex.php?latex=R_%7Bhh%7D%28t_1%2Ct_2%3B%5Ctau_1%2C%5Ctau_2%29%3D+E%5Cleft+%5B+h%28t_1%2C%5Ctau_1%29h%5E%2A%28t_2%2C%5Ctau_2%29+%5Cright+%5D%C2%A0%5Cquad%5Cquad+%282%29+&bg=ffffff&fg=000&s=2&c=20201002)
![R_{hh} (\Delta t ; \tau_1,\tau_2)= E\left[ h(t,\tau_1) h^\ast (t+\Delta t,\tau_2) \right] \quad\quad (3)](https://s0.wp.com/latex.php?latex=R_%7Bhh%7D+%28%5CDelta+t+%3B+%5Ctau_1%2C%5Ctau_2%29%3D+E%5Cleft%5B+h%28t%2C%5Ctau_1%29+h%5E%5Cast+%28t%2B%5CDelta+t%2C%5Ctau_2%29+%5Cright%5D+%5Cquad%5Cquad+%283%29+&bg=ffffff&fg=000&s=2&c=20201002)
![R_{hh}(\Delta t ,\tau )=E\left[ h(t,\tau)h^\ast (t+\Delta t,\tau) \right] \quad\quad (5)](https://s0.wp.com/latex.php?latex=R_%7Bhh%7D%28%5CDelta+t+%2C%5Ctau+%29%3DE%5Cleft%5B+h%28t%2C%5Ctau%29h%5E%5Cast+%28t%2B%5CDelta+t%2C%5Ctau%29+%5Cright%5D+%5Cquad%5Cquad+%285%29+&bg=ffffff&fg=000&s=2&c=20201002)
![p(\tau) = R_{hh}(0,\tau)= E\left [ \left | h(t,\tau) \right | ^2 \right ] \quad\quad (7)](https://s0.wp.com/latex.php?latex=p%28%5Ctau%29+%3D+R_%7Bhh%7D%280%2C%5Ctau%29%3D+E%5Cleft+%5B+%C2%A0%5Cleft+%7C+%C2%A0h%28t%2C%5Ctau%29+%5Cright+%7C+%5E2+%5Cright+%5D%C2%A0%C2%A0+%C2%A0%5Cquad%5Cquad+%287%29+&bg=ffffff&fg=000&s=2&c=20201002)
