IMD3: Third Order Intermodulation Distortion

We’ll begin a discussion on the topic of analog system quality. How do we measure how well an analog system works? One over-simplistic answer is to say that power gain determines how well a system operates. This is not sufficient. Instead, we must analyze the system to determine how well it works as intended, which may include the gain of the fundamental signal. Whether it is an audio amplifier, acoustic transducers, a wireless communication system or optical link, the desired signal (either transmitted or received) needs to be distinguishable from the system noise. Noise, although situationally problematic can usually be averaged out. The presence of other signals are not however. This begs the question, which other signals could we be speaking of, if there is supposed to be only one signal? The answer is that the fundamental signal also comes with second order, third order, fourth order and higher order distortion harmonic and intermodulation signals, which may not be averaged from noise. Consider the following plot:

We usually talk about Third Order Intermodulation Distortion or IMD3 in such systems primarily. Unlike the second and fourth order, the Third Order Intermodulation products are found in the same spectral region as the first order fundamental signals. Second and fourth order distortion can be filtered out using a bandpass filter for the in-band region. Note that the fifth order intermodulation distortion and seventh order intermodulation distortion can also cause an issue in-band, although these signals are usually much weaker.

Consider the use of a radar system. If a return signal is expected in a certain band, we need to be able to distinguish between the actual return and differentiate this from IMD3, else we may not be able to trust our result. We will discuss next how IMD3 is avoided.

Mode Converters and Spot Size Converters

 Spot size converters are important for photonic integrated circuits where a coupling is done between two different waveguide sizes or shapes. The most obvious place to find a spot size converter is between a waveguide of a PIC and a fiber coupling lens.

 Spot size converters feature tapered layers on top of a ridge waveguide for instance, to gradually change the mode while preventing coupling loss.

The below RSoft example shows how an optical path is converted from a more narrow path (such as a waveguide) to a wider path (which could be for a fiber).

While the following simulation is designed in Silicon, similar structures are realized in other platforms such as InP or GaAs/AlGaAs.

RSoft Beamprop simulation, demonstrating conversion between two mode sizes. Optical power loss is calculated in the simulation for the structure.

rsoft13.2

 This is the 3D structure. Notice the red section present carries the more narrower optical path and this section is tapered to a wider path.

rsoft13.1

 The material layers are shown:

rsoft13.3

Structure profile:

rsoft13.5

Discrete-Time Signals and System Properties

First, a comparison between Discrete-Time and Digital signals:

Discrete-TimeDigital
The independent variable (most commonly time) is represented by a sequence of numbers of a fixed interval. Both the independent variable and dependent variable are represented by a sequence of numbers of a fixed interval. 

Discrete-Time and Digital signal examples are shown below:

Discrete-Time Systems and Digital Systems are defined by their inputs and outputs being both either Discrete-Time Signals or Digital Signals.

Discrete-Time Signals

Discrete-Time Signal x[x] is sequence for all integers n.

  Unit Sample Sequence:
๐œน[n]: 1 at n=0, 0 otherwise.  
   Unit Step:
u[n] = 1 at n>=0, 0 otherwise.

Or,

Any sequence: x[n] = a1* ๐œน[n-1] + a2* ๐œน[n-2]+โ€ฆ
where a1, a2 are magnitude at integer n.
                                                or,           

Exponential & Sinusoidal Sequences

Exponential sequence: x[n] = A ๐žชn
                                                 where ๐žช is complex, x[n] = |A|ej๐œ™ |๐žช|ejฯ‰0n=|A||๐žช|n ej(ฯ‰0n+๐œ™)
                                                                                                     = |A||๐žช|n(cos(ฯ‰0n+๐œ™)+j sin(ฯ‰0n+๐œ™))
                Complex and sinusoidal: -๐…< ฯ‰0< ๐… or 0< ฯ‰0< 2๐….

                                                Exponential sequences for given ๐žช (complex ๐žช left, real ๐žช right):

Periodicity:        x[n] = x[n+N],  for all n. (definition). Period = N.
                                Sinusoid: x[n] = A cos(ฯ‰0n+๐œ™) = A cos (ฯ‰0n+ ฯ‰0N+ ๐œ™)
                                                Test: ฯ‰0N = 2๐…k,                            (k is integer)

                                Exponential: x[n] = ejฯ‰0(n+N) = ejฯ‰0n,
                                                Test: ฯ‰0N = 2๐…k,                            (k is integer)

System Properties

                                  System: Applied transformation y[n] = T{x[n]}

Memoryless Systems:

                                Output y[nx] is only dependent on input x[nx] where the same index nx is used for both (no time delay or advance).

Linear Systems:               Adherence to superposition. The additive property and scaling property.

Additive property:         Where y1[n] = T{x1[n]} and y2[n] = T{x2[n]},
y2[n] + y1[n] = T{x1[n]+ x2[n]}.

Scaling property:            T{a.x[n]} = a.y[n]           

Time-Invariant Systems:

                                Time shift of input causes equal time shift of output. T{x[n-M]} = y[n-M]

Causality:

                                The system is causal if output y[n] is only dependent on x[n+M] where M<=0.

Stability:

                                Input x[n] and Output y[n] of system reach a maximum of a number less than infinity. Must hold for all values of n.

Linear Time-Invariant Systems

                                Two Properties: Linear & Time-Invariant follows:

                โ€œResponseโ€ hk[n] describes how system behaves to impulse ๐œน[n-k] occurring at n = k.

  • Convolution Sum: y[n] = x[n]*h[n].

Performing Discrete-Time convolution sum:

  1. Identify bounds of x[k] (where x[k] is non-zero) asN1 and N2.
  2. Determine expression for x[k]h[n-k].
  3. Solve for

General solution for exponential (else use tables):

Graphical solution: superposition of responses hk[n] for corresponding input x[n].

LTI System Properties

As LTI systems are described by convolutionโ€ฆ

LTI is commutative: x[n]*h[n] = h[n]*x[n].

                                โ€ฆ is additive: x[n]*(h1[n]+h2[n]) = x[n]*h1[n] + x[n]*h2[n].

                                โ€ฆ is associative: (x[n]*h1[n])*h2[n] = x[n]*(h1[n]*h2[n])

                LTI is stable if the sum of impulse responses

                                โ€ฆ is causal if h[n] = 0 for n<0                  (causality definition).

Finite-duration Impulse response (FIR) systems:

                Impulse response h[n] has limited non-zero samples. Simple to determine stability (above).

Infinite-duration impulse response (IIR) systems:

                Example: Bh=


If a<1, Bh is stable and (using geom. series) =

Delay on impulse response: h[n] = sequence*delay = (๐œน[n+1]- ๐œน[n])* ๐œน[n-1] = ๐œน[n] – ๐œน[n-1].

______________________________________________________

Continued:

MMIC – A Revolution in Microwave Engineering

One of the most revolutionary inventions in microwave engineering was the MMIC (Monolithic microwave integrated circuits) for high frequency applications. The major advantage of the MMIC was integrating previously bulky components into non-discrete tiny components of a chip. The subsequent image shows the integrated components of the MMIC – spiral inductors (red), FETs (blue) for example.

It is apparent that smaller transistors are present towards the input of the MMIC. This is because less power is required to amplify the weak input signals. As the signals become stronger, higher power (and hence a larger FET) is required. The input terminal (given by the arrow) is the gate and the output the drain. Like almost all RF devices, MMIC’s output and input are usually matched to 50 ohms, making them easy to cascade.

Originally, MMICs found their place within DoD for usage in phased array systems in fighter jets. Today, they are present in cellular phones, which operate in the GHz range much like military RADARs. MMICs have switched from MESFET configurations to HEMTs, which utilize compound semiconductors to create heterostructures. MMICs can be fabricated using Silicon (low cost) or III-V semiconductors which offer higher speed. Additionally, MOSFET transistors are becoming increasingly common due to improved performance over the years. The gate of the MOSFET has been shortened from several microns to several nanometers, allowing better performance at higher frequencies.