Transcription of The Hilbert Transform
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The Hilbert Transform
The Hilbert TransformFrank R. KschischangThe Edward S. Rogers Sr. Departmentof Electrical and Computer EngineeringUniversity of TorontoOctober 22, 20061 DefinitionThe Hilbert transformH[g(t)] of a signalg(t) is defined asH[g(t)] =g(t) 1 t=1 g( )t d =1 g(t ) d .(1)The Hilbert Transform ofg(t) is the convolution ofg(t) with the signal 1/ t. It is the responsetog(t) of a linear time-invariant filter (called a Hilbert transformer) having impulse response1/ t. The Hilbert transformH[g(t)] is often denoted as g(t) or as [g(t)] .A technicality arises immediately. The alert reader will already be concerned with thedefinition (1) as the integral is improper: the integrand has a singularity and the limits ofintegration are infinite.
Some obvious properties of the Hilbert transform follow directly from the definition. Clearly the Hilbert transform of a time-domain signal g(t) is another time-domain signal ˆg(t). If g(t) is real-valued, then so is ˆg(t). Linearity: The Hilbert transform is linear, i.e., if a 1 and a 2 are arbitrary (complex) scalars, and g 1(t) and g
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