# Signal analysis, modeling, processing

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## Transforms; the relation between waveforms and frequency content

...often for converting between time domain and frequency domain.

### Wavelet transforms

For a wider overview, you will want to distinguish between the various related terms, which include (roughly ordered from more general/abstract to more applied/discrete):

• Z-transform
• something of a generalization of the Discrete-time Fourier transform (DTFT, and not to be confused with the DFT)
• Takes a discrete real-numbered or complex-numbered time-domain signal and calculates a complex frequency-domain representation (apparently making it a discrete variation of the Laplace transform)
• Fourier Series
• Usually refers to the idea that x(t) is the sum of an infinite number of sinusoids
• Fourier series integral
• Mathematical basis for Fourier analysis
• Fourier Analysis
• x(t) to {ak} (time series to frequencies)
• (the effective opposite of Fourier Synthesis)
• Fourier Synthesis
• {ak} to x(t) (frequencies to time series)
• (the effective opposite of Fourier Analysis)
• Note that most Fourier transforms have an inverse to allow (re-)synthesis, e.g. iDFT, iFFT, etc.
• Fourier Transform(s)
• Can refer to
• The result of a Fourier analysis (the frequency-domain information)
• The method of converting to that domain, and usually back from its results (Fourier analysis, and synthesis)
• The more mathematical definition of the Continuous Fourier Transform, that transforms a function of a real variable into another, the frequency domain. (in which both are continuous and unbounded(
• A group of implementations for this transform
• The Discrete Fourier Transform (DFT)
• a specific, practicalized version of the (continuous) fourier transform, in that
• it is finite-domain and for discrete-time functions
• input function must be discrete - a real- or complex-numbered time series, often from sampling a continuous signal
• only analyses frequency content required to approximate the given segment (unlike the DTFT)
• Fast Fourier Transform (FFT)
• Refers to a number of different algorithms to quickly calculate a DFT
• ...since the straightforward calculation of DFT takes O(n2), and FFT takes O(n log n)
• Short-term Fourier Transform STFT
• Refers to the fourier transform (often specifically analysis) in which windowing is applied (in the forms of continuous theory, and/or discrete application)
• ...for a controlled tradeoff between frequency resolution and time resolution.

Other related concepts and transforms:

• Discrete-time Fourier transform (DTFT)
• Fractional Fourier transform (FRFT)
• Modified discrete cosine transform (MDCT)
• Modulated complex lapped transform (MCLT)

## Filtering theory

 This article/section is a stub — probably a pile of half-sorted notes, is not well-checked so may have incorrect bits. (Feel free to ignore, fix, or tell me)

## Filters and analyses

 This article/section is a stub — probably a pile of half-sorted notes, is not well-checked so may have incorrect bits. (Feel free to ignore, fix, or tell me)

### Frequency pass and stop filters

#### Low-pass

Filters out frequencies higher than a cutoff frequency.

Regularly used in analog form, e.g. before sampling sound in sound cards, in DSL filters, etc.

Note that blurring samples works as a lowpass filter, in sound, images, and elsewhere.

Anti-aliasing filters are regularly lowpass filters.

#### High-pass

Filters out frequencies lower than a cutoff frequency.

#### Band-stop

Also known as band limit filter, notch filter, 'T-notch filter' and also 'band-elimination filter', and 'band-reject filter'.

The band that is filtered out is referred to as the stopband.

#### Band-pass

The band that is left out is referred to as the passband.

Can be seen as the combination of a high-pass and a low-pass.

### Band-related tools

#### Subband coding

Separate out different frequency bands, to use different coding for each (for reasons of efficient transmission).

### Equalization

Equalization refers to sound processing altering the frequency content/envelope, for correction (e.g. of a recording room's acoustic) or to give sound a particular feel or focus.

Fairly common in software is the graphic equalizer, which adjusts something like 20Hz-20kHz with various sliders, in bands of certain width. There are also hardware equivalents, which may offer feedback detection, to avoid feedback caused by peak sounds (such as vocals) or bad setups (where microphones strongly pick up speaker sound).

Pre-emphasis refers to structurally amplifying some frequencies. For example, vinyl records are mastered with their lower frequencies perhaps 30dB more silent than high frequencies, largely because low frequencies require larger groove size. Mastering with less present low frequencies allows closer grooves and therefore more recording time. Phono pre-amps will equalize the inverse of what was applied to the master, so that they produce approximately what was recorded.

### Comb filter

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A comb filter adds delayed copy of a signal to itself.

### More complex applications

(...which are sometimes presented as complete filters)

#### Noise, SNR, etc.

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Methods for noise reduction / signal (recognition) enhancelent assistance :

• Spectral subtraction (subtracting a pure noise signal's spectrum from te signal's. )
• time domain neural methods (e.g. direct, Kalman)
• frequency domain neural methods
• wavelet shrinkage/thresholding
• ...many more

#### Dynamic range compression

Dynamic range compression (often just 'compression') refers to loudness-adaptive amplification. There are a few different types of uses.

Source-selective compression is usually applied when creating a master for music, to avoid specific instruments from drawing unintended attention, to offset the vocals, for some specific effects such as the (apparent) increased decay time, lessening s sounds in vocals, and other such details.

Overall compression when playing or broadcasting or playing (or, regrettably, mastering).

Applying dynamic range compression on the overall signal is regularly used in broadcasts, to make a station sound both consistent in level and perceptually louder, at expense of some quality (that is, dynamic range). Similarly, home cinema systems often apply this so that you do not have to keep changing the volume in movies with both silent and loud parts.

Record producers have since caught wind of the concept of compression being perceptually louder, and some have forced mastering technicians to apply overall compression, with the idea that louder music is more noticeable and therefore sells better. While the noticability and selling parts may be true, it does not make that much sense to apply this to masters -- radio stations have dynamic range compressors to do exactly the same thing. The net result is that record companies sell music with lowered quality, which will get compressed twice on the radio, on the off chance that a station (and/or DJ) is completely ignorant of these details.

The loudness wars this resulted means that perceptual sparkle mentioned earlier (not easily quantifiable since it is an implication of dynamic range that depends on particular recorded signal/music itself) is reduced, sometimes significantly.