tags: ece-402 dsp spectral-analysis
Short Window Time-Varying Spectral Analysis
Short Window Time-Varying Spectral Analysis is a form of [time-varying spectral analysis] that analyzes only a short segment of a [signal], rather than the entire signal.
- models how the human ear analyzes [frequencies] over short segments of [sound] (about 10-20 ms in length)
- used to perform [spectral analysis] comparable to human hearing
- pitch-synchronous, results in good-time resolution and prevents temporal smearing
- short-window artifacts include cross-talk between frequencies and "ripple" effects
The proper way to extract a short signal segment is to multiply the longer, original signal by a [window function].
Short window analysis is best for monophonic, "quasi-harmonic" input sounds.
- quasi-harmonic: [partial]s of the sound should be very close to integer multiples of the [fundamental]
- for best results, recording should have as little [reverb] as possible
- for the sake of analysis, reverb makes a monophonic sound polyphonic!
Procedure
The following steps are a loose outline of performing short window analysis:
- Detect pitch of [[waveform]] (or ask user)
- Select window, which is related pitch period (usually a two-period window), as well as a hope size
- Compute amplitude and phase at each window (also referred to as "grain")
- String together $A_n(t_0 + m \times N)$ to get amplitude envelope for partial $n$
- Perform frequency correction from phase values
Use of a two-period raised-cosine window:
- all parts of the period are weighted the same
- assume [spectrum] does not change much within the short window
- each window could be thought of as a "grain"
Mathematics
- $s(t)$: input signal to be analyzed, where $t$ is the sample number ($t$ is time in units of $\frac{1}{SR}$)
- $h(i)$: 2-period window
- $N$: input signal's period i samples, rounded to nearest integer
- $f$: frequency in Hz corresponding to period $N$, computed by $f = SR / N$
- $n$: partial number ($n=1$ is fundamental), $1 \leq n \leq N/2$
- $an(t)$: "sine coefficient" for partial $n$ at time $t$, computed using two periods of windowed input data centered at sample $t$ $$ a_n(t) = \sum^N{i=-N}s(i+t) \cdot h(i) \cdot sin(2 \pi nfi / N) $$
- $bn(t)$: "cosine coefficient" for partial $n$ at time $t$, computed using two periods of windowed input data centered at sample $t$ $$ b_n(t) = \sum^N{i=-N}s(i+t) \cdot h(i) \cdot cos(2 \pi nfi / N) $$
- $A_n(t)$: [amplitude] [envelope] value for partial $n$ at time $t$ $$ A_n(t) = \sqrt{a^2_n(t) + b^2_n(t)} $$
- $\theta_n(t)$: phase value for partial $n$ at time $t$ $$ \theta_n(t) = atan(\frac{a_n(t)}{b_n(t)}) $$
- $F_n(t)$: frequency envelope value for partial $n$ at time $t$ $$ F_n(t) = n \cdot (f + \theta_n ' (t)) $$
- $\Delta t$: hop size; usually $\Delta t = N$; $A_n(t)$ and $F_n(t)$ will be computed for $t =$ integer multiples of $t$
- during synthesis, linear interpolation will be done between these computed values of $A_n(t)$ and $F_n(t)$
