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 $R$ is the sample rate).

Formulas:

A sampled real-valued sinusoid looks like:

$$x[n] = a \cdot \cos \left ( \omega n + \phi \right )$$

Here $a$ is the amplitude, $\omega$ the angular frequency, and $\phi$ 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:

$$f = {\omega R \over {2 \pi}}$$

A sampled complex-valued sinusoid looks like:

$$X[n] = A \cdot {{e} ^ { i \omega n }}$$

Or rewritten more simply,

$$X[n] = A {Z^n}$$

where

$$Z = {{e} ^ { i \omega n }} = \cos (\omega n) + i \sin (\omega n)$$

See the Book, section 1.1 (real sinusoids) and then section 7.1 (Time Shifts) for the complex version.

Exercises

1. A complex sinusoid $X[n]$ has frequency 11025 Cycles per second has amplitude 50 and initial phase 135 degrees. Another one, $Y[n]$, has the same frequency, but amplitude 20 and initial phase 45 degrees. What are the amplitude and initial phase of the sum of $X$ and $Y$?
2. What are the frequency, initial phase, and amplitude of the signal obtained when $X[n]$ (above) is delayed 4 samples?
3. What are the frequency, initial phase, and amplitude of the signal product $X[n]Y[n]$?

Sonic challenge

Try to synthesize something like https://msp.ucsd.edu/syllabi/270a.05w/01/crash-stereo.wav.