## Two-Carrier Communication System

Suppose that two Single-Carriers (SC) with symbol periods \(T_1\) and \(T_2\) are amplitude modulated with information \(A\) and \(B\) respectively, the combined signal \(S(t)\) can be described mathematically as in below equation,

## Demodulation Challenges

Requirements by the receiver to extract the information from each of the two sub-carriers are,

- Knowledge of Symbol durations \(\left[T_1, T_2\right]\) (time over which a unique information is transmitted on each sub-carrier).
- Ability to extract information from each sub-carrier without interference from the other.

First, let us try to demodulate the individual information on sub-carriers from the combined signal \(S(t)\) in-order to design \(\left[T_1, T_2\right]\) values. Information on sub-carrier \(T_1\) can be extracted by correlating the combined signal \(S(t)\) with \(\sin \left( \frac{2 \pi t}{T_1} \right)\)

\(S_2(t)_{Intf}\) is the inference from sub-carrier \(T_2\). Similarly, information on sub-carrier \(T_2\) can be extracted by correlating the combined signal \(S(t)\) with \(\sin \left( \frac{2 \pi t}{T_2} \right)\)

\(S_1(t)_{Intf}\) is the inference from sub-carrier \(T_1\). Let us analyse the interference terms \(S_1(t)_{Intf}\) and \(S_2(t)_{Intf}\) to design the values for \(\left[T_1, T_2\right]\). Suppose that \(T_1\) is greater than \(T_2\) then for any positive integer \(n\) we can write \(T_1\) as,

we can re-write the interference terms as,

\(\{B_k\}\) is information transmitted on \(T_2\) carrier in kth symbol duration.

**Integral equation \(S_1(t)_{Intf}\) goes to zero, if \(T_1 = n T_2\) for any positive integer \(n\). So, separation between the sub-carriers in frequency is related by the expression \(T_1 = n T_2\) and the pair \(\left[T_1,T_1/2\right]\) will be the minimum separated orthogonal sub-carriers. Any number orthogonal-carrier system can be designed based on this result.****Integral equation \(S_2(t)_{Intf}\) goes to zero, if all \(\{B_k\}\)'s in \(T_1\) duration are same. So minimum symbol period for a multi-carrier system design is equal to the symbol period of minimum distant sub-carrier frequency from DC.**

## Two-Path Transmission Channel

Transmission medium with multiple delay paths between transmitter and receiver introduce inter-symbol interference (ISI) and also compromise the orthogonality between the sub-carriers in a multi-carrier communication system. Let us analyze a two-tap multi-path channel with channel tap delays \(\left[0,\tau\right]\). For the sake of simplicity let us assume that both channel tap gains are 1, and time domain channel can be described mathematically as \begin{equation} h(t) = \delta(t) + \delta(t - \tau) \end{equation} If the signal described in Figure 3 is passed through channel \(h(t)\), the received signal would be as described in Figure 5, dotted lines describe the signal corresponding to multipath delay \(\tau\).## Resolve Inter-Symbol Interference (ISI)

If the knowledge of maximum delay path is known, mute-periods can be introduced between consecutive data symbol periods to mitigate inter symbol interference. A zero-padded transmission output of a two path multi channel would be as shown in Figure 6 Corresponding recieved signal will be as described in Figure 7 It can be observed that mute-periods (zero-padding) could mitigate ISI but the orthogonality between the sub-carriers is not retained due to multi-path transmission channel. Sub-carrier interference terms due to multi path tranmission can be computed as described below## Retain Sub-carrier Orthogonality

Multi path received signal can retain sub-carrier orthogonality only if the integrals \(S_1^{\tau}(t)_{Intf}\) and \(S_2^{\tau}(t)_{Intf}\) are zero's. This can be achieved if a known signal is transmitted during the mute-periods, that can make the interference integration limits as \(\left[0,T_1\right]\). This problem can be solved if we pre-fix some part of tail of each data symbol to its start, popularly known as**Cyclic Prefix**in

**Orthogonal Frequency Division Multiplexing (OFDM)**system.

Adding Cyclic prefix to the transmitted signal is described in Figure 8 and Figure 9 describes the multipath received signal with cyclic prefix transmission.

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