An exercise often given to students in
introductory engineering circuit laboratory:

The instructor builds a simple
circuit, and hiding it within a black box allows only the input
and output terminals to be exposed. The students must determine
the structure of the circuit and component values by exciting the
circuit with test signals and observing the response.

This procedure is often referred to as "Black Box Identification". The method applies not only to electric circuits, but to any linear or near-linear system including mechanical, chemical, biological, economic, etc. ... In any case, to solve the contents of the black box, one must assume structure and degree of complexity, then test whether measured output data fits the modeled response. If the general structure is known beforehand, the task of identifying parameters is greatly simplified. In practical engineering applications the science of black box identification has greatly advanced with specialized instruments which use signal processing and estimation techniques. These instruments are known as multi-port real time dynamic signal analyzers.

The Dynamic Signal Analyzer evolved from earlier instrumentation, specifically spectrum analyzers and network analyzers. Hewlett Packard [1] makes the following distinction:

**Spectrum Analyzer:**An instrument optimized to characterize signals. Spectrum analyzers introduce very little distortion, and few spurious signals.**Network Analyzer:**An instrument optimized to give accurate amplitude and phase measurements over a wide range of network gains and losses.**Dynamic Signal Analyzer:**An instrument evolved from spectrum analyzer and network analyzer technology providing capabilities of both types of instruments.

- Identification of system transfer functions
- Detection of signal echoes (reflections)
- Detection of periodic signals buried in random noise
- Determination of time delays
- Identification of transmission paths
- Modal analysis of structures
- Characterization of oscillators
- Statistical distributions of signals

Today, spectrum analyzers are most often used for RF and microwave applications, and for the analysis and design of feedback control systems, dynamic signal analyzers are primarily used.

Dynamic signal analyzers typically include one or more types of excitation signal sources within the instrument. These sources are automatically adjusted to excite frequencies within the selected measurement range. Excitation sources are provided on an external port which can be connected to control system inputs.

Although swept sine excitation is often available in these instruments, if the system under measurement is near linear, random excitation is a better choice, as it provides the fastest measurement, providing equal energy across the measurement spectrum concurrent with time.

Measurement processing can be categorized into 4 major steps:

**Recording**Signals are over-sampled at a high rate and processed with an adjustable bandwidth digital decimating filter to prevent aliasing.**Analysis**Recorded signals are processed through FIR windowing filters and an FFT algorithm to produce instantaneous spectra of the signals.**Averaging**Autospectra and cross spectra are calculated and time averaged.**Post Processing**First, the frequency response and coherance function are calculated. From frequency response, autocorrelation, impulse response, cross correlation, and autocorrelation functions are calculated. Results are displayed as Bode, Nyquist, Nichols or impulse response.

Some analyzers provide curve fitting and direct solution of the pole-zero form of the system transfer function. The coherance function is typically used as a weighting function in the estimate. Math functions are often provided allowing manipulation on sets of processed data. One example is the calculation of the open loop transfer function from the closed loop transfer function[2].

Bruel & Kjaer's[3] 2032 and 2034
analyzers provide two built-in methods for calculation of the
complex transfer function. The transfer function, ** H1( f )**,
is defined as the ratio of the cross spectra from input to output
over the input autospectra, and the transfer function,

- For a linear time invariant system,
and in the absence of noise,
and*H1( f )*are equal.*H2( f )* - In the presence of output noise,
provides the best estimate of the transfer function.*H1( f )* - In the presence of input noise,
provides the best estimate of the transfer function.*H2( f )* - For non-linear, and/or noise present
at the input and output of the system,
underestimates the true transfer function, and*H1( f )*overestimates the transfer function. Together*H2( f )*and*H1( f )*define lower and upper bounds of the true transfer function, respectively.*H2( f )* - The ratio of
to*H1( f )*is equal to the coherance function.*H2( f )*

The Coherance function is a measure of how closely the input and output signals are linearly correlated at each frequency. Nearly all dynamic signal analyzers provide coherance function calculation as a means for determining a confidence level for transfer function measurements over the frequency range. A Coherance of 1 represents perfect correlation. Coherance less than 1 at any given frequency can be an indication of any one or more of the following:

- Noise at the input
- Noise at the output
- Non-linearity
- Uncompensated system delays
- Leakage(also known as resolution bias error)

Most dynamic signal analyzers exist as dedicated stand-alone instruments, however some manufacturers provide PC based solutions.

- Tucker Instrument Rentals Lease and purchase of reconditioned analyzers, various manufacturers, includes pricing.
- Ballantine Labs RS 100 series of analyzers, PC based system includes product features and data sheets.
- Signal Processing Systems (SPS) PC based analyzer, description of the SPS390.
- Test Equipment Depot Reconditioned HP analyzers.
- Hewlett Packard Dynamic Signal Analysis Products Complete product information, detailed datasheets.

**[1] The Fundamentals of Signal
Analysis** Application Note 243 Hewlett Packard
Corp.

**[2] Feedback Control System Measurements**
Application Note 240-1 Hewlett Packard Corp.

**[3] Dual Channel FFT Analysis (Part 1)**
Technical Review, A Bruel & Kjaer Publication No. 1, 1984

**[4] Dual Channel FFT Analysis (Part 2)**
Technical Review, A Bruel & Kjaer Publication No. 2, 1984