Fuad Badrieh

A novel iterative method for approximating frequency response with equivalent pole/residues

Frequency domain modeling of interconnects has become the de facto standard for characterization in high-speed signal and power delivery systems. But time domain system level performance analysis calls for pole/residue representation followed by circuit level synthesis from the frequency domain sampled data model of interconnects. In this paper, a new iterative method that produces a set of equivalent poles and residues from discrete sampled frequency response data is proposed. Each iteration picks a few consecutive points from the given sampled response, identifies a local transfer function that matches their response and reduces error by subtracting the local transfer function. Two test cases, strip-line and package, are demonstrated. And results show that the proposed method has potential in fitting system frequency responses and has wide applications in signal and power integrity modeling and simulation of interconnect networks.

Further Examples/Topics on Fourier Transform

With the basic introductory material about the Fourier transform covered in the last two chapters, this chapter drills into some heavy-duty examples including the stair signum function, odd negative exponential, sines/cosines times negative exponential, cropped cosine, cosine squared, cropped t, |t|, t2, and t4 functions, functions of the form f(t)/g(t), arctan function, hat function, tapered pulse, asymmetric triangular function, 3-step stair, truncated pulse train, and the ramped unit step function. Heavy use of applications to drill the idea of the Fourier transform in ones mind and get plenty practice and experience in carrying on the transform. In the chapter we rely heavily on using the Fourier transform properties, developed in the last chapter. We wrap the chapter by demonstrating the flexibility of the Fourier transform by deriving the FT of the unit step function using at least seven methods.

Sampling and the Sampling Theorem

When we do the Fourier or Laplace transform, or when we do their inverse transform we inevitably sample the signal—either in the time domain (former) or frequency domain (latter). The moment we migrate from equations to numbers sampling happens. Question then arises—how fine of a resolution does sampling needs to happen at? As a demonstrate vehicle we assume a time signal of finite bandwidth ω b . We show in the chapter that in order for us to be able to reconstruct the signal back from the Fourier transform the signal must have been sampled at a frequency at least twice the bandwidth frequency ω b . Sampling in the time domain is tantamount of multiplying by the train delta function. The time train function has a Fourier transform which is also a train function, but now in the frequency domain. By making the spacing of the delta functions close enough in the time domain, the spacing in the frequency domain becomes larger. If this spacing is large enough then the spectrum of the sampled signal is repeated (via convolution in the frequency domain) in such a way that each copy of the spectrum which is centered at one of the frequency delta functions does not interfere with the other. As such, doing a low-frequency filtering should enable us to extract the original transform and use to figure the inverse transform (i.e., reconstruct the signal). We show real examples that demonstrate—numerically and graphically—how under-sampling results in problems and over-sampling works perfectly fine.

Causal Cosine and Sine Response

We are all familiar with the sine/cosine system response, especially looking at it from steady state points of view. But the causal sine/cosine response is another story! What the steady state sine/cosine response is missing is the transient part of the solution. But this is not a problem for the causal sine/cosine response; not only do they reproduce the transient solution, they also produce the steady state solution for a net result of capturing the whole solution! Again we start with the transfer function of the system, multiply by either \(\omega _0/(s^2 + \omega _0^2)\) (for sine) or \(s/(s^2+\omega _0^2)\) (for cosine, and where ω0 is the angular frequency of sine/cosine input), then do inverse transform to get the solution in the time domain. Typically multiplying the transfer function by the Laplace transform of the sine/cosine will reproduce the transfer function but with a spike around ω0, and this is discussed in the chapter. Once the response is known in the frequency domain we can find the inverse transform analytically or by numerical integration. We illustrate the causal sine/cosine response on various RLC circuits and study along the way some of the interesting observations about these circuits. Another advantage of knowing the causal sine/cosine response is that we can use them to figure the response to any other causal periodic function simply by using the Fourier series! Keep in mind here we would be using the causal Fourier series, since our basis functions are causal to start with.

Signal Construction in Terms of Convolution Integrals

Two arbitrary signals can be convolved together to get a third signal; but a very relevant and pertinent case is when one of the signals contributes to the input and it is also the output! In other words, how can we generate a signal by convolving it (or a variant thereof) with another? We show in the text how this can be done by various means. For example we can generate the signal by convolving it with the delta function. Or we can generate it by convolving its derivative with the unit step function. Or we can generate it by convolving its second derivative with the ramp function. Or even more we can generate it by convolving its third derivative (divided by 2) with the quadratic function; and so on. But why would we want to go to all this trouble? The premise is, if we know the system (circuit) response to any of the generating functions (the impulse, unit step, …) and if we know how to generate our stimulus of those generating functions (which is the topic of this chapter), then we can know the response of the system due to stimulus by using the same convolution steps—not on the stimulus, but on the system response. For each case we do an example showing all the intermediate details, especially graphically. We wrap the chapter by tying the Fourier transform to the convolution integral using the complex exponential and the delta function.

Fourier Series and Periodic Functions

This is the introductory chapter to spectral methods. The main theme is decomposing a rather arbitrary periodic signal in terms of sines/cosines. The result is the Fourier series and the main task at hand is to figure the expansion coefficients. Those are obtained by integrating the target function against the sine/cosine and give all of DC, sine, and cosine series coefficients. We apply the process on a multitude of signals and stress all the way the visual aspect of the analysis to convince the reader this method works. We see the signal gradually assume the desired shape by including more harmonics. We also learn about the spectrum of the signal which is a plot of the Fourier series coefficients versus frequency. As a reference case we take the periodic pulse and examine its spectrum as a function of period, pulse width, and time elongation and see the corresponding effect on the spectrum. Finally we touch on patching signals and superposition.

Complex Fourier Series

This chapter paves the way for the Fourier transform which most often utilizes complex exponentials for basis functions. It is also the starting point for using complex numbers and analysis in the text. We show in this chapter how to decompose an arbitrary periodic signal in terms of summation of weighted complex exponentials of the form \(f(t) = \sum _n a_n e^{j \omega _n t}\). The expansion coefficients are calculated by integrating the target function against the complex exponentials, for each frequency. We stress that even though the used basis functions are complex they are able to reproduce real functions simply by isolating the real or imaginary parts of the complex exponentials. We also derive the relation between the complex Fourier coefficients to those of the real Fourier series. We demonstrate the process generating the complex Fourier series on a few examples, including the periodic pulse and dwell on the meaning of negative frequencies. We also reexamine the signal spectrum and expand it now to include negative frequencies. The complex Fourier series is the starting point for the complex Fourier transform.

Multi-Port Network: Z- and Y-Parameters

This chapter deals with distributed media with some spatial variations where there are multiple input sources and multiple output observation points. The most common scenario is a 2D conductor slab with multiple ports, where the ports are stimulated by a voltage/current source and the goal is to find the ports current/voltage values, respectively. Putting aside scattering parameters (which are dealt with in the next chapter), the current chapter focuses on three methods: flat RLC grid, impedance (Z-)parameters, and admittance (Y -)parameters. In the flat-grid method we simply discretize the slab into RLC elements, do KVL/KCL for all nodes and branches, and then solve for all voltages and currents. For Z-parameters we derive the impedance matrix by selectively stimulating each port with a current source and observing voltage at all ports. Once the impedance matrix is ready, all that has to be done is to multiply it by any new current vector to give us all port voltages. Similarly for Y -parameters we derive the admittance matrix by selectively stimulating each port with a voltage source and observing current at all ports. Once the admittance matrix is ready, all that has to be done is to multiply it by any new voltage vector to give us all port currents. Of course each of Z-and Y -parameters could be frequency dependent, and hence will build on spectral methods. We wrap the chapter by illustrating the relation between the Z- and Y -matrix.

Unit Step Response as Figured from Inverse Transform

The unit step response of a system is second in order of importance after the impulse response. Before, we figured the unit step response either by directly solving the relevant differential equations, or by simply integrating the impulse response. Here we figure the unit step response by starting with the system transfer function, dividing by s, and then by finding the inverse Laplace transform of the resulting function. The division by s is needed since 1/s is the Laplace transform of the unit step function! Notice that the frequency spectrum of the unit step function samples all frequencies, but more strongly at DC and low frequencies; this is in contrast to the impulse which samples all frequencies uniformly. By doing frequency multiplication the only remaining step is to do the inverse transform (which is frequency integration); but in the impulse response integration approach we have to first do the frequency integration (to get impulse response) and then do time integration (to get step response). We illustrate the unit step response derivation via a handful of examples where we picked the impedance as our transfer function. We apply the method on various RLC networks starting with the transfer function, dividing by s, and then doing inverse transform. Along the way we study some interesting details of those circuits and do some variations on them.

Application to Transistor Modeling and Circuits

The typical introduction to spectral techniques covers linear passive RLC circuits and potentially leaves the impression that active devices, such as MOSFET transistors, work with a different logic. While it is true that active devices are nonlinear, that alone does not preclude them from riding on the spectral techniques wagon! We all have seen the small signal model of transistors, which is an RC network with some controlled sources. Since this simplification, around a DC operating point, results in a passive linear network, the transistor small signal model should abide by the rules and techniques under spectral and convolution umbrella! As such this chapter shows how to extend our analysis of transfer functions to transistors. We start by reviewing the basis behind the large signal operation of the MOSFET to get the IV characteristics and then introduce the small signal model. In the low-frequency version, the small signal model includes only resistors and current sources; in the high-frequency version it also includes parasitic caps. Add to that the load cap and we end up with a multi-branch RC circuit just like we’ve been dealing with all along. We experiment with a few simple MOSFET circuits, derive the operating point, then the small signal model comprised of output impedance and input/output transconductance both of which depend on the MOSFET terminal voltages. Then we derive the system transfer function, such as output voltage versus input one. Knowing the solution in the frequency domain we are assured a time version thereof via inverse transforms.