What is the impulse response of SIMO channel and its equivalent in oversampling?

1- According to the below paragraph :

Could someone explain, What mean M here? I mean when saying with Order M ?

2- On the other hand, When performing Fractional sampling, for example sampling with rate Ts/G (Ts is the sampling rate used at transmitter, and G the oversampling facto), we are expecting to get a discrete-time CIR with GxL channel taps at Ts/G spacing. (exactly as explained in answer below by Zeyad_Zeyad), So the question, is it right that channel response in above equation is similar to SIMO channel response? how ?

Thank you

• Please quote the full paragraph, or the article that contains this sentence. – AlexTP Aug 6 '18 at 17:18
• I don't understand what is "a little bit different, but it's almost the same". No, the equivalent system to a SIMO system is not oversampling the received data by Ts/4. (you probably mean oversampling by 4, right? It's still not equivalent. If it was equivalent, we would less often be doing SIMO/MIMO). And what does all this have to do with the first paragraph? I'm thoroughly confused. Is this about diversity order? Because diversity order is exactly about different observations not being the same as an oversampled single observation. – Marcus Müller Aug 6 '18 at 17:22
• @AlexTP .. OK for example this paper is one of papers which explored that idea with oversampling OR in other words, "Fractional sampling". ieeexplore.ieee.org/document/1344460 .. Using fractional sampling, we can deal with multipath environment as SIMO instead of SISO. As I know, Fractioan sampoing convert SISO into SIMO. So, I think my first part of question is clear, the second part I need to understand, how to observe the channel after taking the FFT in OFDM system, in both SIMO system and SISO with multipath diversity through fractional sampling. . – New_student Aug 7 '18 at 0:40
• @MarcusMüller .. First, thank you for not downvoting :) .. 1- I mean by little bit, that doesn't the same in all cases, because that can be the same in multipath channel, in channels which are not multi-path that's not possible. 2- Ts/4 (where Ts is sampling rate) means oversampling by 4 .. that's right so what's the issue there? 3- the first paragraph, I need to understand (what the impulse response of a SIMO channel with order, for example 16) Then I will compare it into an impulse response of SISO channel with oversampling by 4 and understand the difference .. I think it's clear now .. – New_student Aug 7 '18 at 2:22
• @Eng.Badr I quickly read the IEEE article. Indeed, it is true oversampling helps increasing channel resolution and you can write it down as something similar to SIMO systems. However, they are not the same thing and it is clearly said in the introduction of the article "Since the different channels in the SIMO model resulting from FS are all derived from a single multipath channel, they will be correlated. This limits the amount of diversity obtainable through FS, which we will attempt to quantify, and relate to the transmit signal bandwidth or, equivalently, the pulse shape." – AlexTP Aug 9 '18 at 21:48

Fractional sampling and SIMO are not exactly similar, SIMO system uses multi anntenna at receiver but Fractional sampling can be used with one antenna at receiver. It's said that, Fractional sampling is often used to convert the SISO system into SIMO system without using Multi Antennas at receiver. How?

Suppose a SISO system is used, where you use single Tx antenna with single Rx antenna, Fractional sampling means you oversample the received signal by G, where G is Oversample factor, that means your sampling rate becomes Ts/G, and Ts is the sampling rate used at transmitter. So, that means you don't change the structure of transmitter, but you oversample the receiver. How?

Suppose the received signal is :

where y,u,h are received signal, transmitted signal and channel respectively.

If y(t) is sampled with a rate of Ts/G instead of Ts, where G is the oversampling ratio, and "Ts is the sampling rate which was used at transmitter: its polyphase components can be expressed as: where g = 1, ..., G.

each polyphase component should be demodulated in a separated demodulator (This the one of disadvantage of using Fractional sampling). So instead of having one demodulator, we will have multi demodulators.

Coming back to the previous equation,  and the channel hg[n] is a matrix as below: which means, So now: It seems similar, but I can't confirm or give more details, I am not sure, AlexTP maybe can explain it better than me.

thank you and sorry for the notations, because I don't know how I can write the mathematical equations here.

• Thank you very much .. yes that's what I want to ask about ... I will modify my question and ask again. – New_student Aug 7 '18 at 10:46