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by: Kaya Conroy


Kaya Conroy
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Class Notes
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This 6 page Class Notes was uploaded by Kaya Conroy on Thursday October 22, 2015. The Class Notes belongs to RG ST 7 at University of California Santa Barbara taught by Staff in Fall. Since its upload, it has received 12 views. For similar materials see /class/227182/rg-st-7-university-of-california-santa-barbara in Religious Studies at University of California Santa Barbara.

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Date Created: 10/22/15
Notes 7 ECE594I Fall 2008 BR Brown Electrical Noise Effects SignaltoNoise Ratio and Sensitivity Metrics In this course we are learning how to deal with radiation in terms of the average power transmitted through free space between a target and a sensor We have also shown how to deal with the uctuations of this radiation that occur whether it is incoherent eg thermal or coherent sinusoidal For passive THz systems the radiation propagation through free space is generally handled by the antenna theorem and effective source brightness function For active systems the radiation propagation is generally handled with Friis transmission formula The received power is generally very weak in sensor systems typically orders of magnitude weaker than it is in communications systems So an important issue with any sensor system is masking of the signal by uctuations in the power ie the noise in the receiver The noise is the totality of all the electronic mechanisms in addition to the radiation uctuations Such noise is always present even in the absence of electronic noise so it is important to define a metric for the sensor performance in the presence of noise A useful metric for all types of sensors is the power signal to noise ratio SNR i ltPgt i 1 where Sp is the power spectral density and BN is the equivalent noise bandwidth at that point in the sensor EN Grmx 7 0 where Gf is the sensor gain function vs frequency and Gmax is the maximum value of this gain BN is generally dictated by sensor phenomenology such as the resolution requirements and measurement time Noise from Electronic Components Within every sensor system particularly at the front end are components that contribute significant noise to the detection process and therefore degrade the ultimate detectability of the signal The majority of this noise usually comes from electronics particularly the first device which is often a mixer or direct detector After this there is generally a low level amplifier that contributes comparable noise The majority of noise from such devices falls in two classes 1 Notes 7 ECE594I Fall 2008 BR Brown thermal noise and 2 shot noise Thermal noise in semiconductors is caused by the inevitable uctuations in voltage or current associated with the resistance in and around the active region of the device This causes uctuations in the voltage or current in the device by the same mechanism that causes resistance the Joule heating that couples energy to and from electromagnetic fields The form of the thermal noise is very similar to that for free space blackbody radiation And the Rayleigh Jeans approximation is generally valid for room temperature operation so that the J ohnson Nyquist theorem applies However one must account for the fact that the device is coupled to a transmission line circuit not to a free space mode and the device may not be in equilibrium with the radiation as assumed by the blackbody model All of these issues are addressed by Nyquist s generalized theorem Avs 4kBTDReZDAf12 where TD ZD and Af are the temperature differential impedance and bandwidth of the device Even this generalized form has limitations since it is not straightforward to define the temperature of the device if it is well away from thermal equilibrium Shot noise is a ramification of the device being well out of equilibrium It is generally described as uctuations in the current arriving at the collector or drain of a three terminal device caused by uctuations in the emission time of these same carriers over or through a barrier at or near the emitter or source of the device The mean square current uctuations are given by lt A02 gt 2er1Af where T is a numerical factor for the degree to which the random Poissonian uctuations of emission times is modified by the transport between the emitter or source and collector or drain If T 1 the transport has no effect and the terminal current has the same rms fluctuations When T lt 1 the transport reduces the uctuations usually through some form of degenerative feedback mechanism and the shot noise is said to be suppressed When T gt 1 the transport increases the uctuations usually through some form of regenerative feedback mechanism and the shot noise is said to be enhanced Notes 7 ECE594I Fall 2008 BR Brown Linear Components and Noise Factor While at first appearing to add insurmountable complexity to sensor analysis a great simpli cation results from the fact that radiation noise and two forms of physical noise discussed above are in general statistically Gaussian the shot noise becomes Gaussian in the limit of large samples consistent with the central limit theorem A very important fact is that any Gaussian noise passing through a linear component or network remains statistically Gaussian Hence the output power spectrum S in terms of electrical variable X current or voltage will be white and will satisfy the important identity nA lt AXY gt fSXmdf SXfAf fn where Af is the equivalent noise bandwidth Then one can do circuit and system analysis on noise added by that component at the output port by translating it back to the input port In the language of linear system theory the output and input ports are connected by the system transfer function Hxf so the power spectrum referenced back to the input reference port becomes S X out I HX f I2 39 Because the different Gaussian mechanisms are statistically independent the total noise at SXin the reference point can be written as the uncorrelated sum N N lt Mm2 gt 2lt AX2 gt0r lt me gt lt A392 gt I I As in any RF system it is the signal to noise ratio after detection that matters most And there is always several components between the sensor input and the detector that mask the signal by an amount that depends on the gain ahead of the component This leads to a figure of merit that combines the noise contribution and gain together It is called the noise factor F M 1ltFltoo SN0UT In other words the noise factor quantifies the degradation in SNR as a signal passes through a component in a linear chain It can be combined with the other noise factors in the chain to get F2 1FS 1 Fn l quot71 1 G139 G2 HGi FT0TFi Notes 7 ECE594I Fall 2008 BR Brown where G1 is the power gain of the ih element Intuitively this means that components located further down a chain tend to be less important if the earlier components have high gain Note that this gain accounts for any impedance mismatch between elements so that when the impedance match is poor the gain will suffer too Hence for linear components such as unsaturated ampli ers we have Gf z IHxf2 Physically the noise figure represents the degree to which the linear component degrades the SNR at the input through the introduction of its own noise mechanisms Since linear components generally maintain the statistics of the input noise ie Gaussian noise stays Gaussian the SNR at the output can be no better than the SNR at the input This means that the noise factor can be no less than unity which is why F generally lies in the range 1 lt F lt 00 Note that noise gure which is more commonly used in systems engineering than noise factor is just defined by NF 10log10F System engineers love their decibel units and don t forget that they almost always refer to relative power leveb not signal strength A convenient way to represent the noise figure more intuitively is to assume that the noise at input and output are both additive Gaussian The noise contributed by linear component can then be represented as an equivalent fictitous noise temperature TN corresponding to kBTNBNG noise power at the output or kBTNBN at the input The noise factor is then given by F SNIN j SkBTOBN S N0UT GSGkBT0 kBTNBN After several cancellations we get the simple result F 1 TN T0 where T0 is the ambient temperature Decades ago a convention was established of T0 290 K an average room temperature around the world As an example let s take a modern low noise amplifier LNA which at low enough frequency up to 6 GHz can have a noise gure as low as 10 dB And it can provide this noise f1 gure under room temperature operation The corresponding noise factor and temperature are given by F 10NF10 126 and TN 290F 1 290126 1754 K On first glance this might appear to violate the laws of thermodynamics How can a device produce a noise temperature lower than its operating temperature Fortunately there is no Notes 7 ECE594I Fall 2008 BR Brown violation of thermodynamics because the amplifier is categorically not in thermodynamic equilibrium This is why it can have gain And so consistent with definition the noise temperature is a fictitious measure of just how clean this gain is It is also the reason so many RF and THz engineers work on amplifiers and worry so much about noise figure Quoting an old systems engineer gain is a wonderful thing More General Sensitivity Metrics Unfortunately the noise factor concept is not applicable or particularly useful to all components or THz systems since it presupposes linearity As we shall see shortly there are very good reasons to use nonlinear elements in RF and THz systems not the least of which is the fact that the extraction of information from or making a decision based on the incoming radiation ultimately requires power or energy detection And we know from simple circuit theory that measurement of power or energy is inherently a nonlinear process usually quadratic in the signal levels Hence it is the SNR at the point of detection or decision that is generally the most important quantity in system performance And because of reasons discussed later in statistical detection theory generally this SNR must generally be greater than or equal to unity to have a reliable detection or decision Therefore a very useful metric for sensor performance is to fix the after detection SNR SNRAD at unity and then solve for the signal power at the input to the system sensor that achieves this The resulting metric is the noise equivalent power spectral density NEP which most simply put is the input signal power to the sensor required to achieve a SNR of unity at the output A simple example may be helpful at this point Suppose we have an ideal square law detector which we will address in more detail later because of its great utility in RF and THz systems both A good and very old example is the Schottky diode rectifier By definition the square law detector has a circuit transfer function of Xout mx Pu where Xom is the output signal usually current or voltage Pin is the input power and Six is the responsivity The noise at the output of the detector is minimally given by the Nyquist generalized theorem which for a Schottky rectifier is PN DC ltAI2gt 4kBT0ReYDAf 4kBTOAf RD Notes 7 ECE594I Fall 2008 BR Brown where Af is the post detection bandwidth The corresponding signal power at the output is just X0u2 so that the output SNR with X chosen as I is SNRAD 9 Pn24kBT0Af RD Setting this SNR to unity and solving for Pin we get NEPAD 4kBT0Af RD 12 on For the purpose of comparing different sensor technologies it is conventional to divide out the post detection bandwidth effect or equivalently setting it equal to 1 Hz This yields the normalized NEPAD in units of WHz12 given by NEP4D NEPAD Or NEP AD 4113r1 0RD12mI As a useful example we consider a garden variety Schottky diode used in RF and THz rectification and detection For reasons having to do with the solid state physics of Schottky barriers we find ERI z 25 AW or less And at room temperature RD 1 K92 Choosing these values we get NEP AD z 16x10 13 WHz12 an excellent sensitivity not ever achieved in practice because of power coupling With a differential resistance of 1 KW the power delivery to the device is not very efficient because of impedance mismatch And this problem is pervasive in the THz region where getting adequate device speed and good impedance matching are very difficult to achieve together So we introduce an external power coupling factor 7 analogous to the external coupling efficiency in photonic devices and define it by Pabs 7 Pine where Pine is the power incident on the device or equivalently the available power Then the NEP is re calculated for the incident power that achieves a SNR of unity or NEP AD 4kBT0RD 2n SR1 For the 1 KS2 Schottky diode we expect 7 z 1 R where R is the power re ection coefficient R 1000 RA1000RA2 where RA is the a antenna resistance Choosing a typical value of RA 2 100 S2 we get R 067 7 033 and NEP AD z 5x1013 WHzl2 a value that has been achieved at microwaves but not yet at THz frequencies because the 7 is even lower than 033 largely because of reactive impedance mismatch that we have ignored Nevertheless this example should be illuminating to students trying to understand NEP one of the most bewildering of all system metrics


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