Complex Sinusoid Flashcards
What is the general form of a complex sinusoid?
“y_n = A e^{j (\omega t + \phi)} where A is the amplitude and w is angular frequency and phi is phase.”
“What does Euler’s formula state?”
“Euler’s formula is e^(j0} = cos (0) + j sin(0).”
“How can a complex sinusoid be split into real and imaginary parts?”
“Real: A cos (wt + Ф) Imaginary: A sin(wt + p).”
“What does the real part of a complex sinusoid represent?”
“The real part represents the measurable (physical) signal as cos (wt +Ф).”
What does the imaginary part of a complex sinusoid represent?
“The imaginary part is a mathematical component sin(wt + Ф) used in analysis.”
What does e^{jwt} represent in the complex plane?
“It represents a counterclockwise rotating phasor at angular frequency w.”
“What does en{-jwt} represent?”
“It represents a clockwise-rotating phasor (inverse phasor) at angular frequency w.”
“How is a real sinusoidal signal reconstructed using complex sinusoids?”
“Using the sum of a phasor and its conjugate: cos (wt) = (e^{jut} + e^{-jwt}) / 2.”
What does aliasing mean in the context of real sinusoids?
Aliasing is basically a form of undersampling. The undersampled waveform is constructed to look like a slower frequency waveform or a flat line when the sample rate is the same as the frequency of your signal.
Frequency Modulation
encoding of information in a carrier wave by varying the instantaneous frequency of the wave.
What does aliasing mean in the context of complex sinusoids?
“Aliasing occurs as f_a = f_k - l f s. where l is a positive integer.”
What condition must be satisfied to avoid aliasing?
“The sampling frequency f_s must be at least twice the maximum signal frequency: f_s ≥ 2f_max.
“How does instantaneous frequency relate to the phase of a signal?”
“Instantaneous frequency fra = (1 / 2п) (d phi / dt) where phi is the phase.
What is the discrete approximation for instantaneous frequency?
“f_a ~ (f_s / (Ф_{n+1} (Ф_{n+1} -Ф_n). where Aф is the phase difference between samples.”
“What is a phasor?”
“A phasor is a complex number that represents a sinusoid’s magnitude and phase rotating in the complex plane.
How are complex sinusoids used in Fourier analysis?
“Complex sinusoids form the basis functions for Fourier transforms to analyze signals in the frequency domain.”
What is the difference between a rotating phasor and z^{-1}?
“A rotating phasor (e^{jw}) represents frequency
What does the term z = e^{jw} represent?
“It represents points on the unit circle in the Z-domain corresponding to specific angular frequencies.”
What causes uncertainties in complex sinusoids generation?
Often time-base oscillator because its clock can be inaccurate and depend on physical properties like temperature. This causes clock drifts and can be minimised by resyncing very often like every minute like GPS
What causes uncertainties in complex sinusoids generation?
Often time-base oscillator because its clock can be inaccurate and depend on physical properties like temperature. This causes clock drifts and can be minimised by resyncing very often like every minute like GPS
What are the vector and angle of complex sinusoids?
Vector: amplitude
Angle: phase
