The output noise spectrum of an IC module depends on its input noise and many circuit parameters, such as nominal circuit values, loads, parasitic capacitance, frequency etc. This output noise spectrum could be expressed as a synthetic of the branch noise of the circuit.
Every fault condition can be considered as a change in some branch noise and it will cause a more or less significant change in the output noise spectrum of an IC module. It is available to measure the output noise spectrum and compare the difference between the noise spectrum and that calculated using derivatives.
Noise is generated in any material that is above absolute zero. All electrical devices and their controls also generate noise. Even if there were no external noise, there would still be noise in the output voltage caused by the IC modules themselves. In fact, noise consideration is more complex than that we thought of. We are, in general, still far from having a complete process for the design of an IC module or an electronic system. Although manufacturers supply adequate parameters on manuals for users, they are still faced with some challenging problem like noise estimation of an IC op amp (operational amplifier).
Assuming that the noise characteristics can be somehow determined empirically, a model still need choosing for describing the noise so that it leads to a more accurate fault diagnosis of the IC module. This model is referred to as E_{n}I_{n} noise model, where E_{n} and I_{n} denote equivalent input noise voltage and equivalent input noise current, respectively [6][8]. In this way, all the noise sources in the IC module are assumed to be lumped into a single equivalent input noise voltage and equivalent input noise current. Figure 1 shows the circuit placement of the noise sources in front of a noiseless IC module.
Fig 1: Placement of the equivalent input noise voltage and equivalent input noise current generators in front of a noiseless IC module.

An E_{n}I_{n} model consists of an ideal noiseless amplifier, a noise voltage generator in series with either of the inputs, and a noise current generator in parallel with each input.
What is required is a structured approach to the problem, through which the effects of noise sources can be superposed. If we look at a single element of the ensemble for the time interval 0 £ t £ T we can make up the time average á X(t) ñ of the function X(t)of a random variable x(t) by the definition
 (1) 
If this time average approaches the ensemble average in the limit when T goes to infinity, the noise processes under investigation are said to be ergodic so that the voltage spectral density of a twoport network is given as
 (2) 
Here, denotes the norder random harmonic components of equivalent input noise voltage E(t) , and is the complex conjugate of. Similarly, the current spectral density of an IC network is
 (3) 
The equation is valid if the sufficient condition, is absolutely integrable, is satisfied [9].
Applying (2) and (3) yields
 (4) 
 (5) 
and crossspectral density
 (6) 
Here, Y_{2m} is called shortcircuited forward transfer admittance, something like a transfer function; and H_{2m} is called opencircuited forward transfer admittance. The procedure was described in detail in Literature [7] and [8].
Equation (4) and (5) can be expressed by the matrix relationship existing between outputinput ports of IC op amps
 (7) 
Here,
 (8) 
and
 (9) 
It follows from (7) that the E_{n}I_{n} noise model matrix of an IC op amp is given by
 (10) 
where
 (11) 
and
 (12) 
Here,A^{1} is the inverse matrix of A, and represents the transposed matrix, or Hermitian conjugate matrix [10], of matrix K_{ei}.
These are noise spectral matrices, which are useful if an IC module has to be evaluated repeatedly for several values of the components or the frequency. By a software package, it takes less time to evaluate the same analytical expression with one or a few parameters or the frequency varying over some range than to perform a full analysis each time. It gets easier to perform the analysis with a computer program.
The method was developed for the analysis of rocket designs to identify single point failure modes, that is single component failures, or multistage integrated operational amplifier modes, which fail the system. The analytical expressions do not only allow us to avoid the determination of noise parameter difference between good or bad IC models with timeconsuming, but also enable us to design better systems. The matrix format aids the multilevel analysis of very large systems and generates concise results.
The work presented above is a first step towards the aim, fault location. The second is in development of the practical implementation for ICs to establish the common reasons for failure so that corrective action can be taken.
The output noise spectrum of the system to be detected is derived from those of the noise sources of all the components of the system and their transfer function to the output so that measuring terminals should be determined first. The noise that is contributed by resistors in the circuit should be taken into account. The spectral density of resistor noise is calculated by
 (13) 
where k is Boltzmann's constant, k=1.374´10^{23}(J/K)
and T represents temperature (K).R is the resistor value.
The IC module as shown in Figure 2 provides the example of fault location, the circuit of which is shown in simplified form.
Fig 2: An IC module as an active filter.

The circuit illustrated in Fig.2 can be regarded as a typical twoportnetwork so that its normal output noise spectrum and the tolerance can be obtained by noise matrix approach, i.e. calculate the noise spectrum at the outputs of the IC module to be diagnosed. Details of doing this are given in Literature [11]. In fact, the amplifier noise current spectral density is customarily neglected compared with the noise voltage spectral density [12]. This is justifiable for almost all practical impedance levels associated with both bipolar and MOS realizations[13].
Fig 3: Output noise spectrum of the active filter

During the study, we tested two types of failure sample circuits, circuit 1 and circuit 2. Measured with a FFT analyzer, the two circuits turn out to be faulty circuits. Fig. 3 illustrates the normal noise spectrum, its tolerance (dotted line) and failure conditions of circuit 1 and circuit 2. Comparing with the normal noise spectrum, we could make the decision that the two sample circuit are failures.
The amount of analysis required for a particular system or subsystem should be determined by the accuracy required, the consequences of failure and the nature of the system. The output noise spectrum gives only limited information, and further measurements and calculations for some stages are necessary to locate the failures in the system. Place a short circuit around the input terminals and the input capacitor of the sample circuit 1, and have the circuit from A2 to A3 opened. Measure the output noise spectrum of A1, the first stage of the IC. We found that the measured spectrum is quite different from the spectrum calculated by E_{n}I_{n} model at the normal condition of the system. Hence, the first stage of the fault circuit 1 makes prove to be the failure, which was true.
Diagnosing the fault circuit 2 performs the same procedures as that mentioned above. Placing a short circuit across the output terminals of the first stage of the system, measuring the output noise spectrum of A2, we found that the measured spectrum is quite different from the spectrum calculated by E_{n}I_{n} model at the normal condition of the system. Thus, the second stage of the fault circuit 2 has a fault.
It is advisable to select some internal and external noise measurement points to be used for measurements if some of terminals of the IC are beyond reach. In this case, the output noise spectrum is the sum of the overall noise contribution of some devices in the IC module. The arguments as to whether the correlation coefficient needs to be taken into account for circuit design and simulation still exists. Some researchers like A. V. D. Ziel [9] favor consideration of the correlation. However others, such as Pospieszalski [14], Froelich [15], and Podell [16], do not. In fact, the decomposition of a noise source into two uncorrelated sources greatly simplifies the procedure of predictions of device noise performance, and the quality of the predictions of obtained thereafter.