Problem1) The following set of four waveforms is to be used for transmission across the standard AWGN channel. Assume that b = 6/T s1(t) s2(t) s3(t) s4(t) T T T T T -b (a) Give an orthonormal basis for these waveforms. (b) Sketch the signal space characterization of this set of waveforms, i.e., the signal points Sı; S2; S3; S4. (c) Determine the energy of each of the waveforms.

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Using the orthogonality of sine and cosine makes it possible to transmit andreceive two different signals simultaneously on the same carrier frequency. A scheme to dothis, known as quadrature multiplexing or quadrature amplitude modulation (QAM), is shown in Figure a) Write down the expression for ( ) QAM x t ?b) Derive the two expressions for the outputs 1 y t( ) and 2 y t( ) after the demodulator andlow pass filter.c) Consider the situation when the receiver is not exactly synchronised with thetransmitter, thus the demodulator at the receiver has a slight frequency shiftcos(2 ( ) ) c π f ft + ∆ where ??∆ ≪ ????. Derive the two expressions for the outputs 1 y t( )and 2 y t( ) after the demodulator and low pass filter for this case (Consider only thenoise free situation).

B) A signal f(t) = 2e-atu(t) is applied to the input of system have frequency transfer function Determine the required relations between the constant a and b such that exactly |H(w) = %3D Vw² +b? 50% of the input energy on a 1 ohm basis, is transferred to the output.