Signal Processing

Fourier Transform analysis of sampled sound

Signal Processing is the art and science of modifying acquired time-series data for the purposes of analysis or enhancement. Examples include spectral analysis (using the Fast Fourier or other transforms) and enhancing acquired data using digital filtering.

Igor is ideally suited for signal processing because of its strong support for lengthy time-series (or "waveform") data, and because its many built-in signal processing commands can easily be used through simple dialogs.

In addition, Igor´s programming language makes it straightforward to implement any kind of custom signal processing algorithm, greatly aided by the power of Igor´s Fourier (and other) Transforms.

Fourier Transforms

Igor uses the Fast Fourier Transform (FFT) algorithm to compute a Discrete Fourier Transform (DFT). The FFT can be used to simply characterize the magnitude and phase of a signal, or it can be used in combination with other operations to perform more involved computations such as convolution or correlation.

Spectral Windowing

The FFT computation presumes that the input data repeats over and over. This is important when the initial and final values of your data are not the same: the discontinuity causes aberrations in the spectrum computed by the FFT. "Windowing" smooths the ends of the data to eliminate these aberrations.

Power Spectra

"Power Spectra" answer the question "which frequencies contain the signal´s power?" The answer is in the form of a distribution of power values as a function of frequency, where "power" is considered to be the average of the signal². In the frequency domain, this is the square of FFT´s magnitude.

Power spectra can be computed for the entire signal at once (a "periodogram") or periodograms of segments of the time signal can be averaged together to form the "power spectral density".

Hilbert Transform

The Hilbert Transform computes a time-domain signal that is 90 degrees "out of phase" with the input signal. One-dimensional applications include computing the envelope of a modulated signal and the measurement of the decay rate of an exponentially decaying sinusoid often encountered in underdamped linear and non-linear systems.

Time Frequency Analysis

When you compute the Fourier spectrum (or Power Spectra) of a signal you dispose of all the phase information contained in the Fourier transform. You can find out which frequencies a signal contains but you do not know when these frequencies appear in the signal. For example, consider the signal:

f(x)={sin(2*pi*f1t) 0 ≤ t < t1, sin(2*pi*f2t) t1 ≤ t < t2}

The spectral representation of f(t) remains essentially unchanged if we interchange the two frequencies f1 and f2:

graph showing identical FFT magnitude of two different combinations of sine waves

Clearly the Fourier spectrum is not the best analysis tool for signals whose spectra fluctuate in time.

One solution to this problem is the so-called "Short-time Fourier Transform" (or "Sonogram") in which you can compute the Fourier spectra using a sliding temporal window. By adjusting the width of the window you can determine the time resolution of the resulting spectra.

Other alternative tools are the Wigner transform, the Continuous Wavelet Transform (CWT), and the Discrete Wavelet Transform (DWT).

Convolution and Correlation

You can use convolution to compute the response of a linear system to an input signal. The linear system is defined by its impulse response. The convolution of the input signal and the impulse response is the output signal response. Digital filtering is accomplished by defining a linear system´s impulse response that when convolved with the signal accomplishes the desired result (low-pass or high-pass filter).

The correlation algorithm is very similar mathematically to convolution, but is used for different purposes. It is most frequently used to identify the time delay at which two signals "line up", or are "most similar".

Smoothing

Smoothing removes short-term variations, or "noise" to reveal the important underlying form of the data.

The simplest form of smoothing is the "moving average" which simply replaces each data value with the average of neighboring values. (Other terms for this kind of smoothing are "sliding average", "box smoothing", or "boxcar smoothing".)

Igor´s Smooth operation performs box smoothing, "binomial" (Gaussian) smoothing, and Savitzky-Golay (polynomial) smoothing. The different smoothing algorithms compute weighted averages that multiply neighboring values by differing weights or "coefficients" to compute the smoothed value.

Digital Filters

Digital filters are a natural tool when data is already digitized. Reasons for applying digital filtering to data include:

Elimination of unwanted signal components ("noise")
Enhancing of desired signal components
Detecting the presence of certain signals
Simulation of linear systems (compute the output signal given the input signal and the system´s "transfer function")

Digital filters generally come in two flavors: Finite Impulse Response (FIR) and Infinite Impulse Response (IIR) filters.

Igor implements FIR digital filtering primarily through time-domain convolution using the Smooth or SmoothCustom commands. (In spite of it´s name, SmoothCustom convolves data with user-supplied filter coefficients to implement any kind of FIR filter, low-pass, high-pass, band-pass, etc.)

Design of the FIR filter coefficients used with SmoothCustom is most easily accomplished using the Igor Filter Design Laboratory (a separate product which also requires Igor Pro).

IIR digital filters are designed and applied to data using IFDL.

Level Detection

Level detection is the process of locating the X coordinate at which your data passes through or reaches a given Y value. This is sometimes called "inverse interpolation". Stated another way, level detection answers the question: "given a Y level, what is the corresponding X value?"

Igor provides two kinds of answers to that question. One answer assumes your Y data is a list of unique Y values that increases or decreases monotonically.

The other answer assumes that your Y data varies irregularly, as it would with acquired data. In this case, there may be multiple X values that cross the Y level. Important examples of this are edge and pulse statistics.

A related, but different question is "given a function y = f(x), find x where y is zero (or some other value)". This question is answered by the FindRoots operation.

To detect peaks; see Peak Analysis.

Igor´s Signal Processing Commands

area areaXY BinarySearch
BinarySearchInterp Convolve Correlate
CWT Differentiate DSPDetrend
DSPPeriodogram DWT EdgeStats
faverage faverageXY FFT
FindLevel FindLevels Hanning
HilbertTransform IFFT ImageWindow
Integrate Interp2D Interp3d
interp PCA PulseStats
Rotate Smooth SmoothCustom
Unwrap WignerTransform