INF4420 and

Jørgen Andreas Michaelsen Spring 2013

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Outline

• Noise basics • Component and system noise • Distortion

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INF4420 Spring 2013 Noise and distortion Jørgen Andreas Michaelsen ([email protected]) 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 . E.g. digital switching coupling through to an through the substrate or between metal lines. We mitigate this interference with layout techniques.

True noise is random and zero mean!

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Introduction

TV Static Stock by REC Room

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INF4420 Spring 2013 Noise and distortion Jørgen Andreas Michaelsen ([email protected]) Introduction

We do not know the absolute value, but … • We know the average value • We know the statistical and spectral properties of the noise

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Introduction

• Noise () is always compared to signal (power) • How clearly can we distinguish the signal from the noise (SNR)

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INF4420 Spring 2013 Noise and distortion Jørgen Andreas Michaelsen ([email protected]) Introduction

• Why is noise important? • Signal headroom is reduced when the supply 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.)

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Signal power

driving a dissipates power • The average voltage of the sine wave is zero • The average power dissipated is not zero

2휋 2 2 1 푉 sin 푥 Mean square 퐴 푉 value, RMS is 퐴 푃 = 푑푥 2 2휋 0 푅

1 2휋 푉2 2 2 2 퐴 푣 = 푉퐴 sin 푥 푑푥 = 2휋 0 2

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INF4420 Spring 2013 Noise and distortion Jørgen Andreas Michaelsen ([email protected])

White refers to the spectrum, constant PSD

푉푛 푓 = constant

Resistors exhibit white noise only (ideally). Random thermal motion of (thermal noise)

4푘푇 푉2 푓 = 4푘푇푅 ⇔ 퐼2 = 푅 푅 푅 nV 1 kΩ resistor ≈ 4 @ RT (useful to remember) Hz

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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.

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INF4420 Spring 2013 Noise and distortion Jørgen Andreas Michaelsen ([email protected]) Filtering noise

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Total noise

We find the total noise by integrating from 0 to ∞. This is a finite value because of the filtering.

∞ 1 2 푘푇 4푘푇푅 푑푓 = 0 1 + 푗2휋푓푅퐶 퐶 Resistor 1st order lowpass noise filter magnitude Important! response Total noise is independent of R and defined by C!

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INF4420 Spring 2013 Noise and distortion Jørgen Andreas Michaelsen ([email protected]) Noise bandwidth

Find a , such that when we integrate the ideal constant PSD up to this frequency, the total noise is identical. This is the equivalent noise BW.

휋 휋 1 1 For 1st order filtered noise: 푓 = = 2 푐 2 2휋푅퐶 4푅퐶

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If the noise sources are uncorrelated (common assumption):

2 2 2 푉푛표 = 푉푛1 + 푉푛2

Generally:

2 2 2 푉푛표 = 푉푛1 + 푉푛2 + 2퐶푉푛1푉푛2

퐶 is the correlation coefficient.

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INF4420 Spring 2013 Noise and distortion Jørgen Andreas Michaelsen ([email protected]) Signal to noise ratio

signal power SNR ≡ 10 log dB 10

Example: 푉 푘푇 Best case signal swing, 푉 = 퐷퐷, noise is . 퐴 2 퐶

퐶푉2 SNR = 퐷퐷 8푘푇

If 푉퐷퐷 is reduced, 퐶 must increase: More power (-)

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Sampled noise

When sampling , the sampled spectrum only represents 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 푘푇 still the same, . 퐶

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INF4420 Spring 2013 Noise and distortion Jørgen Andreas Michaelsen ([email protected]) 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.

constant 푉2 푓 = 푛 푓

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Components

exhibit white noise where voltage noise is proportional to resistance. • exhibit white noise () where noise current is proportional to current.

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INF4420 Spring 2013 Noise and distortion Jørgen Andreas Michaelsen ([email protected]) MOSFET noise

White noise (channel resistance) 2 4푘푇 : 퐼푑 푓 = 푟푑푠 γ is process dependent, 2/3 for long-channel 2 2 4푘푇훾 Active: 퐼푑 푓 = 4푘푇훾푔푚 or 푉푔 푓 = 푔푚

Flicker noise (different models are used, we use the one from the textbook) 2 퐾 K is process dependent 푉푔 푓 = 푊퐿퐶표푥 푓

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MOSFET noise

• Total MOSFET noise is white noise +

• Higher 푔푚 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).

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INF4420 Spring 2013 Noise and distortion Jørgen Andreas Michaelsen ([email protected]) Input referred noise

Amplifier not only amplifies the signal, but adds noise. Even though the noise is not physically present at the input, we can calculate an equivalent input noise, 푣푖푛.

푣표푛 푣표 푡 = 4 × 푣푖 푡 + 푣표푛 푡 푣푖푛 = , 퐴0 = 4 퐴0 Amplifier noise

× 4

푣푖 푡 = sin 휔0푡

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Input referred noise

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INF4420 Spring 2013 Noise and distortion Jørgen Andreas Michaelsen ([email protected]) (NF)

A “figure of merit” for how much noise the amplifier adds, compared to the source.

2 4푘푇푅푠 + 푣푎푖 푓 NF 푓 = 10 log10 4푘푇푅푠

Source noise Input referred amplifier noise

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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).

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INF4420 Spring 2013 Noise and distortion Jørgen Andreas Michaelsen ([email protected]) Amplifier system example

Resistive feedback and noiseless opamp 2 푉표 푅2 2 2 푅1 2 = − , 푉푛푖 = 푉푛1 + 푉푛2 푉푖 푅1 푅2 푅2

푅1 푉푖 푉표

Noiseless

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Amplifier system example

푅 2 2 2 2 2 푓 푉푛표1 푓 = 퐼푛1 푓 + 퐼푛푓 푓 + 퐼푛− 푓 1 + 푗2휋푓푅푓퐶푓 2 푅 푅−1 2 2 2 2 2 푓 1 푉푛표2 푓 = 퐼푛+ 푓 푅2 + 푉푛2 푓 + 푉푛 푓 1 + 1 + 푗2휋푓푅푓퐶푓

2 2 푉푛표 푓 = 푉푛표1 푓 2 + 푉푛표2(푓)

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INF4420 Spring 2013 Noise and distortion Jørgen Andreas Michaelsen ([email protected]) Differential pair example

Input stage → most significant noise

Several noise sources, we want the total input referred noise, 2 푉푛푖 푓 , assuming device symmetry.

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Differential pair example

• Ignore 푉푛5, just modulates the bias current • 푉푛1 and 푉푛2 are already at the input • Find how 푉푛3 and 푉푛4 influence the output 2 푔푚3 (푔푚3푅표) and input refer, 푔푚1 2 2 2 2 푔푚3 푉푛푖 푓 = 2푉푛1 푓 + 2푉푛3 푓 × 푔푚1 2 2 훽3 푔푚 = 2훽퐼퐷 → = 2푉푛1 푓 + 2푉푛3 푓 × 훽1 Spring 2013 Noise and distortion 28

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INF4420 Spring 2013 Noise and distortion Jørgen Andreas Michaelsen ([email protected]) Differential pair example

4푘푇훾 Contribution from white noise, 푔푚

1 푔푚3 8푘푇훾 + 8푘푇훾 • Maximize 푔푚1 (small overdrive) 2 • Minimize 푔 (large overdrive) 푔푚1 푔푚1 푚3

퐾 Contribution from flicker noise, 푊퐿퐶표푥푓

2 퐾1 휇푛 퐾3퐿1 • Big devices helps + 2 • Especially increased 푊1 and 퐿3 퐶표푥푓 푊1퐿1 휇푝 푊1퐿3

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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.

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INF4420 Spring 2013 Noise and distortion Jørgen Andreas Michaelsen ([email protected]) Distortion

Amplification and non- linearity depends on the biasing point.

• Soft non-linearity (compression and expansion) • Hard non-linearity (clipping)

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Common source amplifier example

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INF4420 Spring 2013 Noise and distortion Jørgen Andreas Michaelsen ([email protected]) Harmonic distortion

Using Taylor series allows us to study distortion independent of the specific shape of the non- linearity (e.g. common source).

Generic expression for total harmonic distortion (THD). The non-linearity adds harmonics in the frequency domain.

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Harmonic distortion

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INF4420 Spring 2013 Noise and distortion Jørgen Andreas Michaelsen ([email protected]) Harmonic distortion

Using Taylor expansion we write the output of the amplifier as:

2 3 푣표 푡 = 푎 0 + 훼1푣푖 푡 + 훼2푣푖 푡 + 훼3푣푖 푡 + ⋯

Output DC Ideal gain Distortion level (This is the signal we (harmonics) want) (Not important)

In fully differential circuits, the even order terms, 훼2, 훼4, …, cancels out.

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Harmonic distortion

The first terms are the most significant

2 3 푣표 푡 ≈ 훼1푣푖 푡 + 훼2푣푖 푡 + 훼3푣푖 (푡)

In fully differential circuits we approximate the output as

3 푣표 푡 ≈ 훼1푣푖 푡 + 훼3푣푖 (푡)

To analyze the linearity we assume a single tone input: 푣푖 푡 = 퐴 cos 휔푡

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INF4420 Spring 2013 Noise and distortion Jørgen Andreas Michaelsen ([email protected]) Harmonic distortion

From before, 푣푖 푡 = 퐴 cos 휔푡 2 3 푣표 푡 ≈ 훼1푣푖 푡 + 훼2푣푖 푡 + 훼3푣푖 푡

1 + cos 2휃 3 cos 휃 + cos 3휃 cos2 휃 = , cos3 휃 = 2 4

훼 퐴2 푣 푡 ≈ 훼 퐴 cos 휔푡 + 2 1 + cos 2휔푡 표 1 2 3 훼3퐴 + 3 cos 휔푡 + cos 3휔푡 4

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Harmonic distortion

푣표 푡 ≡ 퐻퐷1 cos 휔푡 + 퐻퐷2 cos 2휔푡 + 퐻퐷3 cos 3휔푡

3 퐻 = 훼 퐴 + 훼 퐴3 ≈ 훼 퐴 퐷1 1 4 3 1 훼 퐻 = 2 퐴2 퐷2 2 훼3 퐻 = 퐴3 퐷3 4

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INF4420 Spring 2013 Noise and distortion Jørgen Andreas Michaelsen ([email protected]) Harmonic distortion

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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):

2 2 2 퐻퐷2 + 퐻퐷3 + 퐻퐷4 THD = 10 log10 2 퐻퐷1

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INF4420 Spring 2013 Noise and distortion Jørgen Andreas Michaelsen ([email protected]) Signal to noise and distortion (SNDR)

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Signal to noise and distortion (SNDR)

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INF4420 Spring 2013 Noise and distortion Jørgen Andreas Michaelsen ([email protected]) Third-order intercept point (IP3)

Using two input tones rather than one (same amplitude different frequencies) 푣푖푛 푡 = 퐴 cos 휔1푡 + 퐴 cos 휔2푡 Gives rise to two new distortion components close to the input frequencies 휔1 − Δ휔 and 휔2 + Δ휔 where Δ휔 ≡ 휔2 − 휔1 This distortion increases as 퐴3. We use this to find the intercept point, and infer the third order distortion component.

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Third-order intercept point (IP3)

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INF4420 Spring 2013 Noise and distortion Jørgen Andreas Michaelsen ([email protected]) Spurious free (SFDR)

Ratio between the input signal power and any “spurs” in the spectrum. Could be from harmonics, or feed through from clocks, etc.

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INF4420 Spring 2013 Noise and distortion Jørgen Andreas Michaelsen ([email protected])