a). This signal s(t) leaves the modulator and is sent out across the channel, which adds a noise to the transmitted signal s(t). The resulting signal is r(t)=s(t)+n(t) (8.3) Chapter Eight Channel 8-PSK coder modulator m=(00000000) u=(000 000 000 000) Output symbol # Output symbol Output signal s(t) = Acos(wt) Acos(wt) Acos(wt) Acos(wt) s'(t) 52(t) s (t) s4(t) r(t)=s(t) + n(t) r'(t)=s't)+n'(t) = r*(t)=s*(t)+ n (t) r(t)=r(t)+r2(t)+r(t)+r(t) Figure 8.11 Getting the input to the TCM decoder (a) modulator output s(t) (b) channel output = TCM decoder input = = Trellis-Coded Modulation (TCM): The Wisdom of Modulator and Coder Togetherness which is drawn in Figure 8.11(b). As shown in this figure, another way to express r(t), more convenient in our upcoming presentation, is to write r(t) as a sum of time-sepa- rate components. This leads to an r(t) written as: (8.4) where r1(t)s1(t)+n1(t),r2(t) = s2(t) + n2(t) and SO on. This r(t) feeds the TCM decoder. 8.3.2 The TCM Decoder Front End With r(t) coming in, let's see what the TCM decoder does. It gets started with a decoder front end. Looking at each time interval separately, we see r'(t)=s'(t)+''n (8.5) wheresi(t) Acs(t)(t-(i-1)T) . We know, from our reading in Chapter 5, that the vi(t) can be represented fully on the orthonormal basis (@1)r(t-(i-1)T), Q2(t)= = Specifically, ri(t) can be represented as: (8.6) where (8.7) (8.8) Here, and Ni and ni represent independent Gaussian random variables. Now, Ungerboeck knew this too. He said to himself: Since ri(t) can be fully represented as a vector of two values, what I'll do to start out the construction of the TCM decoder is build a device that maps r((t) into the vector ri = (ri,r2). With that in mind, Ungerboeck drew the TCM decoder front end shown in Figure 8.12. This simply maps ri(t) into its alternative representation of ri = (ri,r21). When the decoder front end was done with the incoming r(t), this is what came out of the receiver front end: IN: r1(t),r2(t),r3(t),r(t) OUT: r1, r2, r3, r4 Chapter Eight r'(t)=s(t)+ n't Q