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.
