fundamentals of dsp part 1 - … · fundamentals of dsp part 1 introduction to digital signal...
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Fundamentals of DSPPart 1Introduction to Digital Signal Processing
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Fundamentals of Digital Signal ProcessingContent
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Part 1: What is a signal?
Time and frequency domain
• Fourier transformation
Part 2: Digitizing signals
• Sampling
• Aliasing
Part 3: Effects to be aware of when converting to digital
• Quantization
• Leakage
Counter-measures to ensure your digital data is valid
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Signals
Analog signal
Signal: measurable quantity carrying information about some physical phenomenon• Pressure, displacement, acceleration, …• Temperature, voltage, biomedical potential (EKG, EEG, ...)
The signal is generated by a structure and detected by a sensor or transducer
• Accelerometer: acceleration voltage• Microphone: pressure voltage • Strain Gauge: strain (deformation) voltage • Thermocouple: temperature changes voltage
The signal is what you want to analyse in view of a particular problem
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Signals
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Signal Processing
Signal processing: specific manipulations of the measured signal to
• Extract key information• Understand physics• Provide input data for specific analysis• Confront simulation results with reality• Modify the signal for specific applications
Signal processing transforms the signal to different domains
• Time domain• Frequency domain• Amplitude domain• Laplace domain• …
Time domain
Frequency domain
Amplitude domain
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Time, frequency and amplitude domains
Each domain = different coordinate system that is used to view or
describe the characteristics of a system or event
Time domain
Frequency domain
Amplitude domain
Each domain highlights a particular aspect of the
characteristics of a system or event
• The time domain is usually the basis for a description of a system’s dynamic behavior. e.g. differential equation of motion. Events are measured as a function of time
• The frequency domain highlights the periodic characteristics of the system or event• The amplitude domain represents looks at the probability distribution of the amplitudes
Transformation
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Fourier transformationJoseph did help us a lot…
Fourier’s law of heat conduction
Analysed in terms of infinite mathematical series
2
2
2
2
y
u
x
uk
t
u-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
-4
-3
-2
-1
0
1
2
3
4
Any signal can be described as a combination of sine waves of different frequencies
Time domain
Frequency domain
Joseph Fourier (º1768 - †1830)
Théorie analytique de
la chaleur (1822)
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Fourier transformation
deXtx
dtetxX
tj
tj
2
1
For mathematicians …
Detect sine waves
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
Frequency domain
Draw line at sine frequency
f [Hz]
• Convert from time to frequency domain and back
• Fourier integral• No information is lost when converting!
-4
-3
-2
-1
0
1
2
3
4
For engineers … Time domain
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Some definitions of sine waves
t [s]T0
Time domain
Period: T0 [s]
f [Hz]
f0
Frequency domain
Frequency: f0 = 1/T0 [Hz]
[rad/s]
0
Pulsation / circular frequency: 0 = 2f0 = 2/T0 [rad/s]
1 rad
2
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Basics of sine waves
f 2
)2sin()( ftAtx
A = Amplitude
f = Frequency
θ = Phase
t = Time
Amp
time
1 secondf is in Hz
ω is in radians/sec
Sine Wave Equation
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Basics of sine wavesFrequency
Amp
time
1 second
Amp
time
1 second
2 Hz 4 Hz
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Basics of sine wavesAmplitude
Amp
time
1 second
1 g 0.5 g
Amp
time
1 second
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Basics of sine wavesAmplitude
Amp
time
5 g Peak
0
-5
5
3.5 g RMS (.707 of Peak)
10 g Peak-to-Peak (2xPeak)
Scaling can cause amplitude difference!
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Basics of sine wavesPhase
)2sin()( ftAtx
Phase is measured as an angleOrange signal “lags” the green by about 45° or /4 radians Green signal “leads” the orange by about 315° or 7/4 radians
Phase is the amount of shift in time relative to another reference (another signal, start of FFT block, etc.)
Amp
time0
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Fourier transformation
A: A spectrum in the frequency domain. It represents a series of sines and cosines in the form of complex numbers. When these numbers are summed, they form the original signal in the time domain.
Solution:
Note: You will get a complex number for each point in your spectrum
a + jb = complex number
a = real part
b = imaginary part
j = -1
Q: What is the output of a Fourier transformation?
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Frequency spectrumComplex numbers
Hz
Log
Phas
e
A: Spectrum most commonly viewed as Magnitude/Phase
Magnitude
Phase
Q: How do I make sense of this complex numbo-jumbo?
AKA: Bode Plot
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Frequency spectrumTime history
Selection of domain depends on the application aimsEquivalence of time and frequency domain: no loss of information
Time TimeFrequency Frequencyf
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Signals
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PART 2: Digitizing signals
• Sampling
• Aliasing
PART 3: Effects to be aware of when converting to digital
• Quantization
• Leakage
Counter-measures to ensure your digital data is valid
To be continued….
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Fundamentals of DSPPart 2Introduction to Digital Signal Processing
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Fundamentals of Digital Signal ProcessingContent
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Part 1: What is a signal?
Time and frequency domain
• Fourier transformation
Part 2: Digitizing signals
• Sampling
• Aliasing
Part 3: Effects to be aware of when converting to digital
• Quantization
• Leakage
Counter-measures to ensure your digital data is valid
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Nice theory…but we need to do this on a computer
Digital Signal Processing: apply manipulations using a computer-based system
• Most transducers output an analog (continuous) signal• Computers are digital devices (0/1; on/off)• Convert the sensor signal into a discrete stream of digital
information• Discretization in time and in amplitude• Massive loss of information when sampling
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Sampling analog signals
10 Hz sine wave, sampled at 512 Hz: digital representation looks like a perfect sine
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Sampling analog signals
10 Hz sine wave, sampled at 128 Hz: digital representation still looks OK
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Sampling analog signals
10 Hz sine wave, sampled at 64 Hz: digital representation starts looking strange…
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Sampling analog signals
10 Hz sine wave sampled high (512 Hz, red) and low (64 Hz, green) on a digital oscilloscope
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Sampling analog signalsExploring the limits…
Sampling frequency = sine wave frequencyfs = fsine
Sampling frequency = 2 x sine wave frequencyfs = 2 x fsine
Observed frequency = 0 Hz (DC) Observed frequency is correct, but it is borderline (sampling frequency cannot be lowered)
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Sampling analog signalsWhen pushing further, you get aliasing
Sine wave frequency = 20 HzSample rate 21.3 Hz
Observed frequency is wrong: 20 Hz sine wave sampled at 21.3 shows as 1.3 Hz signal
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Sampling analog signalsWhen pushing further, you get aliasing
2max
sff
Nyquist frequency
fs 2fs 3fsfs/20
True
frequencies
“Observed” frequencies
fs/2
Aliasing
20 Frequency
CorrectAmplitude
0 1.3
Observed
Usable frequency
range
Information to be discardedNo usable data
21.3
Sample rate
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Sampling analog signalsHow to prevent aliasing?
Select sample rate to cover full signal bandwidthIf there is no frequency content above Nyquist Frequency, then there is no AliasingThis is not always practical or possible:
• Large files sizes • Limitations of data acquisition equipment
Anti-aliasing filter Ideal case
Ampl
itude
Frequency fs/2=fmax
Low pass filter to prevent aliasing:Removes frequencies that violate Nyquist from analog signal before digitizing
0
1
Limit signal bandwidth using Anti-Aliasing Filter• Analog and/or Digital low-pass filters
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Be aware that the amplitude of the last portion of the spectrum is
• attenuated by the anti-alias filter• contaminated by out-of-band signal
Limit signal bandwidth using Anti-Aliasing Filter• Analog and/or Digital low-pass filters
Anti-aliasing filter Ideal case
Ampl
itude
Frequency fs/2=fmax
Low pass filter to prevent aliasing:Removes frequencies that violate Nyquist from analog signal before digitizing
0
1
Sampling analog signalsHow to prevent aliasing?
Make sure the signal does not contain frequencies above half the sample frequency
Extremely sharp analog filter, but “brick wall” effect is not possibleRoll off point starts @ 80% of bandwidth
fs/2=fmax0
1
80% of fmax
Ampl
itude
Anti-aliasing filter practical
Frequency
Do this by applying a sufficient performing low-pass filter
Automatically done in good data acquisition hardware
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Aliasing demonstrationSine sweep from 45 to 82 Hz, sampled at 128 Hz
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PART 3: Effects to be aware of when converting to digital
• Quantization
• Leakage
Counter-measures to ensure your digital data is valid
To be continued….
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Fundamentals of DSPPart 3Introduction to Digital Signal Processing
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Part 1: What is a signal?
Time and frequency domain
• Fourier transformation
Part 2: Digitizing signals
• Sampling
• Aliasing
Part 3: Effects to be aware of when converting to digital
• Quantization
• Leakage
Counter-measures to ensure your digital data is valid
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Digitizing analog signalsQuantization resolution
Sampling in time domain:Store an amplitude value on a periodic basis (fs)
Amplitude is determined with a discrete resolution
Real values are rounded or truncated to discrete levels
This process causes quantization noise
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Digitizing analog signalsSome formulas
• Assume ADC with M bits at the output• The number of voltage intervals N is given by
𝑁 = 2𝑀 − 1
• The voltage resolution of an ADC is equal to its overall voltage measurement range divided by the number of discrete values:
∆𝑉 =𝑉𝑟𝑎𝑛𝑔𝑒
2𝑀
• The Signal-to-quantization-noise ratio is given by
𝑆𝑄𝑁𝑅 = 20 𝑙𝑜𝑔10 2𝑀 ≈ 6.02 ∗ 𝑀 𝑑𝐵
M N ΔV SQNR# of bits # of voltage
stepsVoltage
resolution(for +/- 10V range)
Signal to quantification
noise ratio4 16 1.24 V 24 dB8 255 78.1 mV 48 dB
12 4 095 4.88 mV 72 dB16 65 535 0.305 mV 96 dB24 16 777 215 1.19 µV 144 dB
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Quantization demonstration
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Quantization demonstration
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Quantization demonstration
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Mitigating quantization errorsSome terminology
Analog Digital
Time
Input rangeor
Full scale rangeor
ADC range
- V
+ V Headroom or Overhead
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Mitigating quantization errorsDemonstration: 10 Hz, 2 mV sine wave
16 bit ADC, 10 V input range 0.305 mV discretization level
2 mV signal / 0.305 mV = 13 truncated to 13 possible levels
2 possible solutions:
Use ADC with higher number of bits
Apply gain to the signal prior to ADC
Large quantization errors!
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Select the right range,but watch out for overloads!
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We don’t have all day…Finite observation period
Periodic observation: correct amplitude level at correct spectral line
Fourier
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If signal is not periodic in observation window Leakage
Periodic observation: correct amplitude level at correct spectral line
Fourier
Leakage = severe distortion of spectrum if the signal is not periodic in the observation window
A-periodic observation: up to 63% amplitude error at spectral line closest to correct frequency
Remaining 37% amplitude spread out over entire frequency spectrum
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Leakage demonstration
Slow sweep from 10 to 11 Hz, showing spectrum with 0.5H resolution
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Effect of finite observation time
Leakage when sine wave frequency falls between the spectral lines
Leakage when signal is not periodic within the observation window
Sine frequency 10.7 Hz
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Why is this happening?Periodicity assumption
Observation window is assumed to repeat itself, introducing ‘spikes’ in the signal
Fourier
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Reduce the effect of leakage
‘Force’ the input for the Fourier transformation to be periodically expandable
Practical implementation: multiply signal with time domain window to eliminate discontinuities
Fourier
Effects of time window:
Improved amplitude estimate flatten central lobe
Reduce frequency range of smearing lower side lobes
Local smearing of spectral energy due to wider central lobe lower effective spectral resolution
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Window typesmost popular ones
Tim
e do
mai
nFr
eq. d
omai
nRectangular, uniform Hanning Flat top
AKA
No Window
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