**Summary**

- Introduction
- Geometric representation of M energy signals
- Review of the digital memoryless modulation techniques

The **Gram-Schmidt procedure** allows us to represent a set of M signal waveforms of duration T with a set of N < M functions Ψ_{k}(t) of a basis with duration T (k=1,..,N)

The r. v. ni e nk are uncorrelated and, since Gaussian are also statistically independent.

Given an arbitrary complete basis the componentsof a white signal are uncorrelated.

If the signal is Gaussian the components will be also indipendent.

Since all the signals s_{m}(t) are represented by a single basis function (namely, Ψ(t)), the PAM signalling is monodimensional (**N=1**).

The MPSK signalling can be obtained with two modulators PAM with carriers in quadrature.

The components must verify the requirement over the energy ⇒ They are not **independent**.

*5*. Digital Transmission over AWGN chanel

*6*. Evaluation of P(e) for optimum RX in AWGN

*7*. Error probability for M-PSK

*8*. Noncoherent ML Demodulation of FSK signals in AWGN

*9*. Transmission through bandlimited AWGN channels

*13*. Cyclic Codes

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