Notes #7a, ECE594I, Fall 2009, E.R. Brown 121 Noise effects and signal-to-noise ratio In this course we have seen 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 fluctuations of this radiation that occur whether it is incoherent (e.g., thermal) or coherent (sinusoidal). For passive RF 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 fluctuations in the power (i.e., the “noise”) in the receiver. The “noise” is the totality of all the electronic mechanisms, in addition to the radiation fluctuations. 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). ENBP2BSP)P(PNS⋅><=><><=∆ (1) where SP is the power spectral density and BENB is the equivalent noise bandwidth at that point in the sensor ∫∞−=01max)()( dffGGBENB where G(f) is the sensor gain function vs frequency, and Gmax is the maximum value of this gain BENB 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) thermal noise, and (2) shot noise. Thermal noise in semiconductors is caused by the inevitableNotes #7a, ECE594I, Fall 2009, E.R. Brown 122 fluctuations in voltage or current associated with the resistance in and around the active region of the device. This causes fluctuations 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 Johnson-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 ∆Vrms = [4kBTDRe{ZD}∆f ]1/2 where TD, ZD , and ∆f 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 fluctuations in the current arriving at the collector (or drain) of a three-terminal device caused by fluctuations 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 fluctuations are given by fIei ∆⋅Γ>=∆< 2)(2 where Γ is a numerical factor for the degree to which the random Poissonian fluctuations of emission times is modified by the transport between the emitter (or source) and collector (or drain). If Γ = 1, the transport has no effect and the terminal current has the same rms fluctuations. When Γ < 1, the transport reduces the fluctuations, usually through some form of degenerative feedback mechanism, and the shot noise is said to be suppressed. When Γ > 1, the transport increases the fluctuations, usually through some form of regenerative feedback mechanism, and the shot noise is said to be enhanced.Notes #7a, ECE594I, Fall 2009, E.R. Brown 123 Linear Components and Noise Factor While at first appearing to add insurmountable complexity to sensor analysis, a great simplification 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 ffSdffSXXfffX∆⋅≈⋅>=∆<∫∆+)()()(002 where ∆f 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 HX(f), so the power spectrum referenced back to the input (reference) port becomes 2|)(|)()(fHoutSinSXXX=. Because the different Gaussian mechanisms are statistically independent, the total noise at the reference point can be written as the uncorrelated sum ∑=>∆<>=∆<NjiTOTXX122)()( or ∑=>∆<>=∆<NjiTOTPP122)()( 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 figure (or factor) ⎟⎟⎠⎞⎜⎜⎝⎛=OUTINNSNSF)/()/( ; ∞<<F1 In other words, the noise figure quantifies the degradation in SNR as a signal passes through a component in a linear chain. It can be combined with the other noise figures in the chain to getNotes #7a, ECE594I,