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Music 170: Formula sheet and Problem Set #1
(due Jan. 13)
Conventions:
I'll use capitals for complex numbers, lower-case for real numbers, and
greek letters for anything in radians. (Until I start forgetting. Note the
first exception below where is the sample rate).
Formulas:
A sampled real-valued sinusoid looks like:
Here is the amplitude, the angular frequency, and
the initial phase. The angular frequency is in radians per
sample, and the initial phase in radians. The frequency can be
expressed in cycles per second as:
A sampled complex-valued sinusoid looks like:
Or rewritten more simply,
where
See the Book, section 1.1 (real sinusoids) and then section 7.1 (Time Shifts)
for the complex version.
Exercises
- A complex sinusoid has frequency 11025 Cycles per second has amplitude
50 and initial phase 135 degrees. Another one, , has the same frequency,
but amplitude 20 and initial phase 45 degrees. What are the amplitude and
initial phase of the sum of and ?
- What are the frequency, initial phase, and amplitude of the signal
obtained when (above) is delayed 4 samples?
- What are the frequency, initial phase, and amplitude of the signal
product ?
Sonic challenge
Try to synthesize something like
https://msp.ucsd.edu/syllabi/270a.05w/01/crash-stereo.wav.
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Miller Puckette
2005-01-06