Outline. Noise and Distortion. Noise basics Component and system noise Distortion INF4420. Jørgen Andreas Michaelsen Spring / 45 2 / 45
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1 INF440 Noise and Distortion Jørgen Andreas Michaelsen Spring / 45 Outline Noise basics Component and system noise Distortion Spring 013 Noise and distortion / 45
2 Introduction We have already considered one type of noise in the layout lecture: Interference Other circuits or other parts of the circuit interfering with the signal. E.g. digital switching coupling through to an analog signal through the substrate or capacitive coupling between metal lines. We mitigate this interference with layout techniques. True noise is random and zero mean! Spring 013 Noise and distortion 3 3 / 45 Introduction TV Static Stock Video by REC Room Spring 013 Noise and distortion 4 4 / 45
3 Introduction We do not know the absolute value, but We know the average value We know the statistical and spectral properties of the noise Spring 013 Noise and distortion 5 5 / 45 Introduction Noise (power) is always compared to signal (power) How clearly can we distinguish the signal from the noise (SNR) Spring 013 Noise and distortion 6 6 / 45
4 Introduction Why is noise important? Signal headroom is reduced when the supply voltage is reduced. Supply voltage is reduced because of scaling (beneficial for digital, less power consumption) Noise is constant. Noise limits performance of our circuits (along with mismatch, etc.) Spring 013 Noise and distortion 7 7 / 45 Signal power Sine wave driving a resistor dissipates power The average voltage of the sine wave is zero The average power dissipated is not zero P = 1 π v = 1 π 0 π 0 π V A sin x R dx V A sin x dx = V A Mean square value, RMS is V A Spring 013 Noise and distortion 8 8 / 45
5 White noise White refers to the spectrum, constant PSD V n f = constant Resistors exhibit white noise only (ideally). Random thermal motion of electrons (thermal noise) V R f = 4kTR I R = 4kT R 1 kω resistor 4 RT (useful to remember) Hz Spring 013 Noise and distortion 9 9 / 45 Filtering noise In most circuits we have capacitors (noise free) There is always some parasitic capacitance present. White noise is filtered: the constant PSD is shaped. Spring 013 Noise and distortion / 45
6 Filtering noise Spring 013 Noise and distortion / 45 Total noise We find the total noise by integrating from 0 to. This is a finite value because of the filtering jπfrc 1 st order lowpass filter magnitude response 4kTR df = kt C Resistor noise Important! Total noise is independent of R and defined by C! Spring 013 Noise and distortion 1 1 / 45
7 Noise bandwidth Find a frequency, such that when we integrate the ideal constant PSD up to this frequency, the total noise is identical. This is the equivalent noise BW. For 1 st order filtered noise: π f c = π 1 = 1 πrc 4RC Spring 013 Noise and distortion / 45 Adding noise sources If the noise sources are uncorrelated (common assumption): V no Generally: V no = V n1 + V n = V n1 + V n + CV n1 V n C is the correlation coefficient. Spring 013 Noise and distortion / 45
8 Signal to noise ratio SNR 10 log 10 signal power noise power db Example: Best case signal swing, V A = V DD kt, noise is. C SNR = CV DD 8kT If V DD is reduced, C must increase: More power (-) Spring 013 Noise and distortion / 45 Sampled noise When sampling signals, the sampled spectrum only represents frequencies up to the Nyquist frequency (half the sampling frequency). Frequencies beyond are aliased down to this range. (More on this in a later lecture). The total noise is still the same, kt C. Spring 013 Noise and distortion / 45
9 1/f noise So far, we have considered white noise (with filtering). Transistors exhibit significant flicker (1/f) noise. The PSD is proportional to the inverse of the frequency. V n f = constant f Spring 013 Noise and distortion / 45 Components Resistors exhibit white noise where voltage noise is proportional to resistance. Diodes exhibit white noise (shot noise) where noise current is proportional to diode current. Spring 013 Noise and distortion / 45
10 MOSFET noise White noise (channel resistance) Triode: I d f = 4kT r ds Active: I d f = 4kTγg m or V g f = 4kTγ g m Flicker noise (different models are used, we use the one from the textbook) V g f = K WLC ox f γ is process dependent, /3 for long-channel K is process dependent Spring 013 Noise and distortion / 45 MOSFET noise Total MOSFET noise is white noise + flicker noise Higher g m attenuates the white noise contribution Larger devices (gate area) reduces the flicker noise We can reduce flicker noise using circuit techniques. E.g. in sampled analog systems we can sample the noise and subtract (correlated double sampling). Spring 013 Noise and distortion 0 0 / 45
11 Input referred noise Amplifier not only amplifies the signal, but adds noise. Even though the amplifier noise is not physically present at the input, we can calculate an equivalent input noise, v in. v in = v on A 0, A 0 = 4 4 v o t = 4 v i t + v on t Amplifier noise v i t = sin ω 0 t Spring 013 Noise and distortion 1 1 / 45 Input referred noise Spring 013 Noise and distortion / 45
12 Noise figure (NF) A figure of merit for how much noise the amplifier adds, compared to the source. 4kTR s + v ai f NF f = 10 log 10 4kTR s Source noise Input referred amplifier noise Spring 013 Noise and distortion 3 3 / 45 Input referred noise The amplifier may be a cascade of gain stages. High gain at the input stage attenuates noise contributed by later stages (important). Spring 013 Noise and distortion 4 4 / 45
13 Amplifier system example Resistive feedback and noiseless opamp V o = R, V V i R ni = V n1 + R 1 1 R R V n V i R 1 V o Noiseless Spring 013 Noise and distortion 5 5 / 45 Amplifier system example V no1 f = I n1 f + I nf f + I n f V no f = I n+ f R + V n f + V n f 1 + R f 1 + jπfr f C f 1 R f R jπfr f C f f V no f = V no1 + V no (f) Spring 013 Noise and distortion 6 6 / 45
14 Differential pair example Input stage most significant noise Several noise sources, we want the total input referred noise, V ni f, assuming device symmetry. Spring 013 Noise and distortion 7 7 / 45 Differential pair example Ignore V n5, just modulates the bias current V n1 and V n are already at the input Find how V n3 and V n4 influence the output g m = (g m3 R o ) and input refer, g m3 g m1 V ni f = V n1 f + V n3 f g m3 g m1 βi D = V n1 f + V n3 f β 3 β 1 Spring 013 Noise and distortion 8 8 / 45
15 Differential pair example Contribution from white noise, 4kTγ g m 8kTγ 1 + 8kTγ g m3 g m1 g m1 Maximize g m1 (small overdrive) Minimize g m3 (large overdrive) Contribution from flicker noise, K WLC ox f C ox f K 1 W 1 L 1 + μ n μ p K 3 L 1 W 1 L 3 Big devices helps Especially increased W 1 and L 3 Spring 013 Noise and distortion 9 9 / 45 Distortion So far we have discussed noise, which is an important performance constraint The degree of non-linearity is another important performance limitation SNR improves with stronger input signals (because the noise remains constant) However, larger input amplitude adds more non-linearity. The sum of noise and distortion is important (SNDR). Increasing the amplitude improves SNDR up to some point where the non-linearity becomes significant. Beyond this point the SNDR deteriorates. Spring 013 Noise and distortion / 45
16 Distortion Amplification and nonlinearity depends on the biasing point. Soft non-linearity (compression and expansion) Hard non-linearity (clipping) Spring 013 Noise and distortion / 45 Common source amplifier example Spring 013 Noise and distortion 3 3 / 45
17 Harmonic distortion Using Taylor series allows us to study distortion independent of the specific shape of the nonlinearity (e.g. common source). Generic expression for total harmonic distortion (THD). The non-linearity adds harmonics in the frequency domain. Spring 013 Noise and distortion / 45 Harmonic distortion Spring 013 Noise and distortion / 45
18 Harmonic distortion Using Taylor expansion we write the output of the amplifier as: v o t = a 0 + α 1 v i t + α v i t + α 3 v i 3 t + Output DC level (Not important) Ideal gain (This is the signal we want) Distortion (harmonics) In fully differential circuits, the even order terms, α, α 4,, cancels out. Spring 013 Noise and distortion / 45 Harmonic distortion The first terms are the most significant v o t α 1 v i t + α v i t + α 3 v i 3 (t) In fully differential circuits we approximate the output as v o t α 1 v i t + α 3 v i 3 (t) To analyze the linearity we assume a single tone input: v i t = A cos ωt Spring 013 Noise and distortion / 45
19 Harmonic distortion From before, v i t = A cos ωt v o t α 1 v i t + α v i t + α 3 v i 3 t cos θ = 1 + cos θ, cos 3 θ = v o t α 1 A cos ωt + α A + α 3A cos ωt + cos 3ωt 3 cos θ + cos 3θ cos ωt Spring 013 Noise and distortion / 45 Harmonic distortion v o t H D1 cos ωt + H D cos ωt + H D3 cos 3ωt H D1 = α 1 A α 3A 3 α 1 A H D = α A H D3 = α 3 4 A3 Spring 013 Noise and distortion / 45
20 Harmonic distortion Spring 013 Noise and distortion / 45 Total harmonic distortion As with noise, the ratio between the distortion and the signal is what we are interested in. Total harmonic distortion (sum of all harmonics relative to the fundamental tone): H D + H D3 + H D4 THD = 10 log 10 H D1 Spring 013 Noise and distortion / 45
21 Signal to noise and distortion (SNDR) Spring 013 Noise and distortion / 45 Signal to noise and distortion (SNDR) Spring 013 Noise and distortion 4 4 / 45
22 Third-order intercept point (IP3) Using two input tones rather than one (same amplitude different frequencies) v in t = A cos ω 1 t + A cos ω t Gives rise to two new distortion components close to the input frequencies ω 1 Δω and ω + Δω where Δω ω ω 1 This distortion increases as A 3. We use this to find the intercept point, and infer the third order distortion component. Spring 013 Noise and distortion / 45 Third-order intercept point (IP3) Spring 013 Noise and distortion / 45
23 Spurious free dynamic range (SFDR) Ratio between the input signal power and any spurs in the spectrum. Could be from harmonics, or feed through from clocks, etc. Spring 013 Noise and distortion / 45
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