A2.4: Complex Representation of Signals

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A2.4 Complex Representation of Signals Systems

• Topics
• Pre-envelope.
• Canonical representations of band-pass signals.
• Band-pass systems.

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PRE-ENVELOPE

• The pre-envelope or the analytical signal of the
  signal g(t) is defined as

where is the Hilbert transform of g(t).

• Note that the given signal g(t) is the real part
  of the pre-envelope and the Hilbert
  transform of the signal is the imaginary part of
  the pre-envelope.

• Phasors simplify manipulations of alternating
  currents and voltages similarly, the pre-envelope
  is useful in handling band-pass signals and
  systems.

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PRE-ENVELOPE

• The Fourier transform of the pre-envelope can be
  written as

from the above expression one can easily conclude
that

• It is important to note that the pre-envelope of
  a signal has no frequency contents for all
  negative frequencies.

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PRE-ENVELOPE

• For a given signal g(t), the pre-envelope can be
  determined by using one of the two methods

• First determine the hilbert transform of the
  signal g(t) and use equation (A2.36) to determine
  the pre-envelope

• Determine the FT of the signal g(t) and use
  equation (A2.37) to determine and the
  evaluate the inverse FT to obtain

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PRE-ENVELOPE

• The pre-envelope for negative frequencies is
  defined as

• The two pre-envelopes are complex conjugates of
  each other

• The FT of the pre-envelope for negative
  frequencies is

• The pre-envelope is non-zero for
  negative frequencies.

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Canonical Representation of Bandpass Signals

• The pre-envelope of a narrowband signal g(t) with
  FT G(f) centered about some frequency can
  be expressed as

where is called the complex envelope
which is a low pass signal.

• By definition the real part of the pre-envelope
  is the given signal

• In general is complex-valued that can be
  expressed in the following form

where both and are real-valued
low-pass functions.
where is called the complex envelope which
is a low pass signal
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Canonical Representation of Bandpass Signals

• The original bandpass signal g(t) can be
  expressed in the standard or canonical form as

with and referred to as the
in-phase and quadrature component of the signal.

• Some Formulas

The envelope (magnitude of compex envelope)
in-phase component
Phase
quadrature component
where is called the complex envelope which
is a low pass signal
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Bandpass System

• A bandpass system with impulse response h(t) can
  be expressed as

• The complex impulse response of the bandpass
  system can be written as

• It can easily be shown that

• Taking the FT we have

using the fact that
for a real impulse response and that there is no
overlapping between and we have
for positive frequencies
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Evaluating the Response of a Bandpass System

• The bandpass input signal x(t) is obtained by its
  complex envelope which is related to the input as

• The bandpass system with impulse response h(t) is
  obtained by its complex envelope which is related
  to h(t) as

• The complex envelope of the output bandpass
  signal y(t) is obtained by using

• Finally, the desired output can be obtained from
  the complex envelope by using the following
  relation