Introduction to Signal Integrity

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Introduction to Signal Integrity

Outline

• Module 1: Signals and basics of Fourier Transform. (TM)
• Module 2: Pulse or Tone. (IM)
• Module 3: Numerical Computation of Fourier Transform. (TM)
• Module 4: Case Study. (IM)
• Module 5: Errors in Computing Fourier Transform (TM/IM).
• Module 6: Exit Ticket, Takeaways.

Module 1

Signals and basics of Fourier Transform

Signals

• SI: Integrity of Signals.
• As humans: A signal is a change in time.
• As systems: A signal is a sum (interference) of harmonics.
• As SI Engineer: Fluent in both languages!

Signals: Humans Perspective

• Humans think of signals as values at instances.
• Each "instant" (0,0,0...,1,0,…) is a coordinate axis.
• A signal is a vector. Projection on axis is strength of the signal.

Signals: Systems Perspective

• Systems "think" of signals as complex amplitudes of harmonics.
• Each "harmonic " exp(i2πft) is a "coordinate axis".
• A signal is a vector. Projection on "harmonic" axis is amplitude of the signal.

Signals: SI Perspective

• Signals are vectors
• Creatures exist irrelevant to how we describe them.
• Belong to a high dimensional vector space (conceptual).

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Fourier Transforms are change of coordinates

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Fourier Analysis in a nutshell

• Represents signals in terms of "harmonics".
• For an SI engineer: explore signals from a different angle.

Where do we use Fourier Transform?

• Everywhere!

Engineering

Medical

Physics

Chemistry

Earth Science

Computer Vision

What is common between all these applications?

What is common between all these applications?

Signals

Questions:

• As an SI Engineer, do I really care about having an understanding of Fourier Analysis?

Questions:

• As an SI Engineer, do I really care about having an understanding of Fourier Analysis?

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• Fourier transform is summarized in three letters "FFT". Shouldn't this mature and robust tool take care of all "issues"?

Welcome to the world of harmonics

Harmonics addition may appear counter-intuitive

https://www.reddit.com/r/EngineeringStudents/comments/dr3f59/fourier_transform_bad/

Fourier Transform as rotation of coordinates

Signal Vector represented in time coordinates

FT: Change coordinates to harmonics

Same Signal Vector represented in harmonics coordinates

Apply System

IFT: Change coordinates to time

Fourier Transforms

Time

Frequency

Simple Example

Simple Example

Fourier Transform (rotate coordinates)

Simple Example

Fourier Transform (rotate coordinates)

Note: Vector does not change. Coordinates do!

Continuous Transform

Time and Frequency are continuous

Dual (FT and IFT look the same)

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What is the meaning of the Negative frequency?

Two interesting Properties

Shift Property

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Uncertainty Property

Translate in time equivalent to phase shift in frequency

(longer interconnects, extra phase)

Wider bandwidth, shorter  duration

Shift property

time

freq

Shift property

Implication in SI

• Generate pulses sequences.

Shift property

Implication in SI

• Generate pulses sequences.
• Basics of some Equalizers (FFE, DFE).

Uncertainty property

• Inverse relation between signal width in time and frequency.
• Bandlimited signals go from -∞ to ∞ in time.
• Time limited signals go from -∞ to ∞ in frequency.
• Notation borrowed from Quantum mechanics.
• Extreme cases:
• Impulse: Defined instant of time, undefined frequency.
• Sinusoid: Defined frequency, undefined time.

Implication of Uncertainty principle

Module 2

Pulse or Tone

Old days

Pulse or Tone?

Pulse Dialing

Module 3

Numerical Computation of Fourier Transform

Not one Fourier Transform

Time

Frequency

CFT/ICFT

DFT/IDFT

Continuous Time/Discrete Frequency

Discrete Time/Continuous Frequency

Not one Fourier Transform

Time

Frequency

CFT/ICFT

(Physical Signals)

DFT/IDFT

Continuous Time/Discrete Frequency

Discrete Time/Continuous Frequency

Not one Fourier Transform

Time

Frequency

CFT/ICFT

(Physical Signals)

DFT/IDFT

Continuous Time/Discrete Frequency

(S parameters)

Discrete Time/Continuous Frequency

Not one Fourier Transform

Time

Frequency

CFT/ICFT

(Physical Signals)

DFT/IDFT

Continuous Time/Discrete Frequency

(S parameters)

Discrete Time/Continuous Frequency

(Time Sampled signals)

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Not one Fourier Transform

Time

Frequency

CFT/ICFT

(Physical Signals)

DFT/IDFT

(Computed Signals)

Continuous Time/Discrete Frequency

(S parameters)

Discrete Time/Continuous Frequency

(Time Sampled signals)

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Discrete Fourier Transform

This is what numerical tools compute using FFT

Mathematically Inclined: Visualizing IDFT in terms of Vectors

Mathematically Inclined: Visualizing IDFT in terms of Vectors

Mathematically Inclined: Visualizing IDFT in terms of Vectors

Mathematically Inclined: Visualizing IDFT in terms of Vectors

Projections

Harmonics

Computing CFT and ICFT

Note that

• Actual Signals moving around are continuous.
• Continuous Fourier Transform (CFT) and Inverse Continuous Fourier Transform are most relevant.
• We want to leverage the computational efficiency of FFT.

Our Task:

Compute CFT using DFT

Basic Idea:

Assignment 1

• Will be posted online

Module 4

Errors in Computing Fourier Transform

Sampling issues

Errors in CFT

Errors in CFT

Spectrum of a Sampled signal

Spectrum of a Sampled signal

Spectrum of a Sampled signal

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Animation: Spectrum of a Sampled signal

Truncation Error (Spectral Leakage)

• S parameters measurements
• Truncation is equivalent to convolution with sinc in time domain.

Truncation Error (Spectral Leakage)

• S parameters measurements
• Truncation is equivalent to convolution with sinc in time domain.

Wide

Medium

Narrow

What about frequency sampled signals?

• Using Duality:

So for a bandlimited signal as long as Fs>2fmax no error?

Why?

So for a bandlimited signal as long as Fs>2fmax no error?​

Module 5

Case Study

Case Study

• Consider the following: You want to characterize two cables of lengths 1ft and 3ft respectively. Assume that the 1ft cable delay is 3 ns per ft and its loss is about –15 dB per ft at 15 GHz. For the 1ft cable:
• What is the effect of measuring the cable up to 5 GHz, 10 GHz, 50 GHz?
• Estimate the resolution frequency to use in measurement.
• How the values would change (if any) when you characterize the 3ft cable?

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Effect of maximum frequency

• Truncation error
• Change maximum frequency from 100 GHz down to 5 GHz

Effect of resolution frequency

• Aliasing error
• Change Delay from 0 to 0.5 ns

Module 6

Exit Ticket

Reflect on what we discussed today (5mins)

• Discuss with your Seatmate.

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Takeaways Points

• Time and Frequency representations are complementary.
• Fourier transform important properties: shift and uncertainty and implications in SI.
• FFT efficient way to compute DFT which can be used to approximate CFT.
• There are inevitably errors in computing CFT.
• Two types of errors:
• Aliasing.
• Truncation.