# The Hilbert Transform - Part 1

This is part one of a multi-part series that defines The Hilbert Transform and its significance. 1. Definition The Hilbert transform of a signal is defined as \begin{eqnarray} \mathscr{H}[g(t)] = g(t) * \frac{1}{\pi t} = \frac{1}{\pi} \int_{-\infty}^{\infty} \frac{g(\tau)}{t-\tau} d\tau = \frac{1}{\pi} \int_{-\infty}^{\infty} \frac{g(t-\tau)}{\tau} d\tau \end{eqnarray} The Hilbert transform of is the convolution of with the signal […]

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# Know Your Eb/N0 - Part 3

This is the last part of our three-part series that explains the meaning of in digital communication context. Calculating Having understood how Gaussian noise comes to affect transmitted symbols, it remains to define an appropriate average signal energy, and hence a signal-to-noise ratio (SNR). A commonly used definition of SNR is , where is the […]

# Know Your Eb/N0 - Part 2

This is the second part of our three-part series that explains the meaning of in digital communication context. Signal Constellations Often, particularly in the specification of signal sets used in modems, we rely on the notion of an orthonormal basis. A set is an orthonormal basis if   The linear span of consists of the […]

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# Know Your Eb/N0 - Part 1

This is part one of a three-part series that explains the meaning of in digital communication context. Introduction Every student of communications knows that digital communication system performance—for example, bit error rate (BER) or frame error rate (FER)—depends on the signal-to-noise ratio (SNR), with higher SNRs leading to lower BERs and FERs. Sometimes less clear, […]

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