The book Fundamentals of Wireless Communication had shown that the input/output model of the wireless system we usually studied exists in discrete-time complex baseband. Typically, the narrowband model is
y=Hx+n
where H is the discrete-time complex baseband channel which is obtained by sampling the convolution of baseband equivalent input and baseband equivalent impulse response.
However, I find that in many papers they directly use the Fourier transformation of the baseband equivalent impulse reponse as H [1-3]. Others directly use the parametric channel model [4-5] as H. It appears that there is inconsistency about the use of channel model. I wonder why they can directly use the Fourier transformation of the baseband equivalent impulse reponse or parametric channel model as the discrete-time complex baseband channel H.
[1] Heath, Robert W., et al. "An overview of signal processing techniques for millimeter wave MIMO systems." IEEE journal of selected topics in signal processing 10.3 (2016): 436-453.
[2] Sayeed, Akbar M., et al. "Wireless communication and sensing in multipath environments using multi-antenna transceivers." Handbook on Array Processing and Sensor Networks. Wiley, 2010. 115-170.
[3] Wang, Bolei, et al. "Spatial-and frequency-wideband effects in millimeter-wave massive MIMO systems." IEEE Transactions on Signal Processing 66.13 (2018): 3393-3406.
[4] Abbas, Waqas Bin, Felipe Gomez-Cuba, and Michele Zorzi. "Millimeter wave receiver efficiency: A comprehensive comparison of beamforming schemes with low resolution ADCs." IEEE Transactions on Wireless Communications 16.12 (2017): 8131-8146.
[5] Alkhateeb, Ahmed. "DeepMIMO: A generic deep learning dataset for millimeter wave and massive MIMO applications." arXiv preprint arXiv:1902.06435 (2019).
I didn't find any literature elaborating the relationship between baseband frequency response and discrete-time baseband channel. It seems people just use it without reason. That's very confusing.
I have tried to explain the steps that are needed to go from a continuous-time wideband channel to a discrete-time channel model using OFDM on page 4-10 in the following paper: https://arxiv.org/pdf/2102.00742.pdf
There is also an appendix that discusses some of these things in further detail: https://github.com/emilbjornson/SP_Cup_2021/blob/main/appendix_Ref1.pdf
The paper is about reconfigurable intelligent surfaces, but the basic theory is the same.
Dear Mengyuan Ma ,
I will give a conceptual answer to your question.
The channel description follows the rules of the description of the linear systems.
So the channel can be considered in continuous time domain t by its impulse response h(t).
In this case the the relation between the output and input of the channel in time domain has the form:
y(t) = h(t) * x(t)
where * is the convolution process.
In digital systems where the signal are sampled an discretized according to the sampling theorem the signal will be discrete in time in form of a sequence in time with sampling frequency fs=1/Ts with Ts is the sampling time.
In this case the above relation becomes:
y[n] = h[n] * x[n]
which is the discrete time nTs form.
The third from is the description of the channel characteristics at the continuous frequency domain f
The channel frequency response will be H(f) which is the Fourier Transform of h(t)
The above relation in frequency domain turns out to be
Y = H(f) X,
The form of the signals will be complex having real and real and imaginary parts.
H(f) is the frequency response of the channel.
In the discrete frequency response is the discretization of the continuous frequency response.
The domain which sums up the two domains is the complex frequency S-domain
such that La Place h(t) = H(s)
H(s) can be converted to the frequency response by just substituting s= jw then
H(s) will be transformed to H(f).
Also h[n] the discrete time function can be transformed into discrete time La place transformation Z such that h(n) will be transformed into H(z).
where Z= exp (sTs)
In the discrete frequency domain,
Z= exp jwTs
So, finally in the discrete frequency domain the channel description will be
H( exp jwts) instead of H(jw)= H(f) in continuous time and frequency domains.
In summary, the channel is considered as a linear system and its description follow the linear systems as I explained before.
Best wishes
It seems that only when the system is OFDM, can we directly use the frequency response of baseband CIR as channel H, because the input/output model of OFDM is in frequency domain.
The use of Fourier transform of baseband equivalent impulse response or parametric channel model as the discrete-time complex baseband channel H is mainly due to the assumption of wideband channel models. In such cases, the channel impulse response is modeled as a sum of complex exponentials that extend over a wide bandwidth. In this case, it is more convenient to represent the channel in the frequency domain, where the frequency response can be expressed as a product of the transmitted signal and the channel frequency response. This is known as the frequency-selective fading model and is commonly used in wideband wireless communication systems.
In contrast, the narrowband model assumes that the signal bandwidth is much smaller than the channel coherence bandwidth, and the channel impulse response can be well approximated by a discrete-time complex baseband channel model. In this case, the channel impulse response is obtained by sampling the continuous-time impulse response. The resulting channel model is expressed in terms of baseband equivalent channel impulse response and is generally used in narrowband wireless communication systems.
So, the choice of channel model mainly depends on the nature of the wireless channel and the bandwidth of the transmitted signal. For narrowband channels, the discrete-time complex baseband model is more appropriate, while for wideband channels, the frequency-selective fading model or parametric channel model may be used.
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