©Themoon EXAM SOLUTIONS
26/11/2024 10:59AM
ECE 2026 Digital Signal Processing All
Solutions
DSP - Answers✓✓Digital Signal Processing
ECE 2026 - Answers✓✓The biggest weed out class for electrical and computer engineers at
Georgia Institute of Technology.
Sinusoidal Signals - Answers✓✓a mathematical curve that describes a smooth repetitive
oscillation. It is named after the function sine, of which it is the graph. It occurs often in pure
and applied mathematics, as well as physics, engineering, signal processing and many other
fields. Its most basic form as a function of time (t) is:
y(t) = A* sin(2\π * ft + α )
Amplitude, Phase, and Frequency - Answers✓✓Amplitude - the maximum extent of a vibration
or oscillation, measured from the position of equilibrium.
Phase - One is the initial angle of a sinusoidal function at its origin and is sometimes called
phase offset or phase difference. Another usage is the fraction of the wave cycle that has
elapsed relative to the origin.
Frequency - frequency is defined as a number of cycles per unit time. This unit of time often is
called the Period(T). So F = 1/T.
, ©Themoon EXAM SOLUTIONS
26/11/2024 10:59AM
Complex Exponential Representation (Phasors) - Answers✓✓In physics and engineering, a
phasor (a portmanteau of phase vector[1][2]), is a complex number representing a sinusoidal
function whose amplitude (A), angular frequency (ω), and initial phase (θ) are time-invariant. It
is related to a more general concept called analytic representation,[3] which decomposes a
sinusoid into the product of a complex constant and a factor that encapsulates the frequency and
time dependence. The complex constant, which encapsulates amplitude and phase dependence,
is known as phasor, complex amplitude, and (in older texts) sinor, or even complexor.
So the phasor in a sinusoidal function is the non-time-variant real part.
sinusoid = A*cos(wt+θ) = Re{ Ae^i(wt+θ)} =
Re{ Ae^iθ *e^iwt }
Ae^iθ is the phasor
e^iwt is the time-variant part of the sinusoid
Refer to Wiki:
https://en.wikipedia.org/wiki/Phasor
Multiplication of a Phasor by a Scalar - Answers✓✓Multiplication of the phasor {Ae^iθ} by a
complex constant, {Be^iФ}, produces a different phasor. That means its only effect is to
change the amplitude and phase of the underlying sinusoid.
Re{ (Ae^iθ*Be^iФ) * e^iwt } =
Re{ AB*e^i(θ+Ф) * e^iwt} = ABcos(wt+(θ+Ф))
Differentiation of a Phasor - Answers✓✓The time derivative or integral of a phasor produces a
different phasor. That means its only effect is to change the amplitude and phase of the
underlying sinusoid.
26/11/2024 10:59AM
ECE 2026 Digital Signal Processing All
Solutions
DSP - Answers✓✓Digital Signal Processing
ECE 2026 - Answers✓✓The biggest weed out class for electrical and computer engineers at
Georgia Institute of Technology.
Sinusoidal Signals - Answers✓✓a mathematical curve that describes a smooth repetitive
oscillation. It is named after the function sine, of which it is the graph. It occurs often in pure
and applied mathematics, as well as physics, engineering, signal processing and many other
fields. Its most basic form as a function of time (t) is:
y(t) = A* sin(2\π * ft + α )
Amplitude, Phase, and Frequency - Answers✓✓Amplitude - the maximum extent of a vibration
or oscillation, measured from the position of equilibrium.
Phase - One is the initial angle of a sinusoidal function at its origin and is sometimes called
phase offset or phase difference. Another usage is the fraction of the wave cycle that has
elapsed relative to the origin.
Frequency - frequency is defined as a number of cycles per unit time. This unit of time often is
called the Period(T). So F = 1/T.
, ©Themoon EXAM SOLUTIONS
26/11/2024 10:59AM
Complex Exponential Representation (Phasors) - Answers✓✓In physics and engineering, a
phasor (a portmanteau of phase vector[1][2]), is a complex number representing a sinusoidal
function whose amplitude (A), angular frequency (ω), and initial phase (θ) are time-invariant. It
is related to a more general concept called analytic representation,[3] which decomposes a
sinusoid into the product of a complex constant and a factor that encapsulates the frequency and
time dependence. The complex constant, which encapsulates amplitude and phase dependence,
is known as phasor, complex amplitude, and (in older texts) sinor, or even complexor.
So the phasor in a sinusoidal function is the non-time-variant real part.
sinusoid = A*cos(wt+θ) = Re{ Ae^i(wt+θ)} =
Re{ Ae^iθ *e^iwt }
Ae^iθ is the phasor
e^iwt is the time-variant part of the sinusoid
Refer to Wiki:
https://en.wikipedia.org/wiki/Phasor
Multiplication of a Phasor by a Scalar - Answers✓✓Multiplication of the phasor {Ae^iθ} by a
complex constant, {Be^iФ}, produces a different phasor. That means its only effect is to
change the amplitude and phase of the underlying sinusoid.
Re{ (Ae^iθ*Be^iФ) * e^iwt } =
Re{ AB*e^i(θ+Ф) * e^iwt} = ABcos(wt+(θ+Ф))
Differentiation of a Phasor - Answers✓✓The time derivative or integral of a phasor produces a
different phasor. That means its only effect is to change the amplitude and phase of the
underlying sinusoid.
