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Data Analysis: Time and Frequency Domain
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Typical Data Acquisition System
SignalSource
SignalConnection
SignalConditioning
+-
SignalMeasurement
ADC
Analog
Digital
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DigitizationDigitization
• An analog signal is sampled at a point in time and converted to a time series
• An analog signal is sampled at a point in time and converted to a time series
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DigitizationDigitization
• Each sampled signal value is digitized using and analog-to-digital converter
• Parameters:– Resolution: number of bits used to represent the
analog signal– Range: min. and max. voltage ADC can span (-5V
to +5V)– Gain: range scale factor (gain factor of 10 means
that a range spans 1/10 of the original range).– Polarity: single (-5 to 5V) or double (0 to 10V)
• Each sampled signal value is digitized using and analog-to-digital converter
• Parameters:– Resolution: number of bits used to represent the
analog signal– Range: min. and max. voltage ADC can span (-5V
to +5V)– Gain: range scale factor (gain factor of 10 means
that a range spans 1/10 of the original range).– Polarity: single (-5 to 5V) or double (0 to 10V)
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Code Width (LSB)Code Width (LSB)
• Number of codes is a function of resolution:
#of codes = 2
• Code width (vertical sensitivity is the amount of voltage corresponding to a one-bit increment in code number
LSB =
• Number of codes is a function of resolution:
#of codes = 2
• Code width (vertical sensitivity is the amount of voltage corresponding to a one-bit increment in code number
LSB =
resolution
range
gain x #of codes
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Code Value to VoltageCode Value to Voltage
• Conversion :
voltage = (code) x code_width +
• Conversion :
voltage = (code) x code_width + Bottom of range
gain
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When to Sample?When to Sample?
• Settling time is important• Settling time is important
desiredmeasured
measureddesired
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When to Sample?When to Sample?
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Improperly sampledImproperly sampled Properly sampledProperly sampled
fN = fs/2 fs: sampling frequencyfN = fs/2 fs: sampling frequency
Sampling GuidelinesSampling Guidelines• Nyquist Theorem
sampling rate > 2 x maximum frequency of signal
• Nyquist Frequency (fN)
maximum frequency that can be analyzed
• Frequencies above Nyquist Frequency cause aliasing
• Nyquist Theorem sampling rate > 2 x maximum frequency of signal
• Nyquist Frequency (fN)
maximum frequency that can be analyzed
• Frequencies above Nyquist Frequency cause aliasing
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What is Aliasing? (Time Domain)What is Aliasing? (Time Domain)
• Samples acquired at 1 kHz• Samples acquired at 1 kHz
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150 Hz
150 Hz sine tone ? 150 Hz sine tone ?
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150 Hz
850 Hz
850 Hz sine tone ? (1000 Hz – 150 Hz) 850 Hz sine tone ? (1000 Hz – 150 Hz)
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Acquired
150 Hz
850 Hz
1150 Hz
1150 Hz sine tone ? (1000 Hz + 150 Hz) 1150 Hz sine tone ? (1000 Hz + 150 Hz)
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850 Hz
1150 Hz
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150 Hz
850 Hz
1150 Hz
... Hz
n * Fsampling 150 Hz n * Fsampling 150 Hz
Aliasing (Frequency Domain)Aliasing (Frequency Domain)
• 150, 850, and 1150 Hz• 150, 850, and 1150 Hz
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f1f1 f3f3
fs /2fs /2 fsfs
alias free bandwidthalias free bandwidth
f1f1
fs /2fs /2 fsfs
anti-aliasingfilteranti-aliasingfilter
f2f2
attenuatedf2
attenuatedf2
aliasf3
aliasf3
f4f4
RAW SIGNALRAW SIGNAL
ACQUIRED SIGNALACQUIRED SIGNAL
Time Domain ConsiderationsAlias Free BandwidthTime Domain ConsiderationsAlias Free Bandwidth Nyquist FrequencyNyquist Frequency Sample FrequencySample Frequency
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• Removes frequencies higher than Nyquist frequency
• Analog low-pass filter
• Before sampling
• Removes frequencies higher than Nyquist frequency
• Analog low-pass filter
• Before sampling
Time Domain ConsiderationsAnti-Aliasing FilterTime Domain ConsiderationsAnti-Aliasing Filter
Flat FrequencyResponse
SharpRoll-off
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850 Hz
1150 Hz
... Hz
AA-Filter
Anti-Aliasing Filter (Analog Only)Anti-Aliasing Filter (Analog Only)
Analog anti-aliasing filter – Passband – DC to 400 Hz– Stopband – 600 Hz
Analog anti-aliasing filter – Passband – DC to 400 Hz– Stopband – 600 Hz
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Sampling MethodsSampling Methods
• Simultaneous Sampling
• Interval Sampling
• Continuous Sampling
• Random Sampling
• Multiplexing
• Simultaneous Sampling
• Interval Sampling
• Continuous Sampling
• Random Sampling
• Multiplexing
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Simultaneous SamplingSimultaneous Sampling
• Critical time relation btw. signals
• Requires:– Sample-and-hold circuits OR– Individual ADC’s
• Critical time relation btw. signals
• Requires:– Sample-and-hold circuits OR– Individual ADC’s
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Interval SamplingInterval Sampling
• Simulate simultaneous sampling for low-frequency signals
• Simulate simultaneous sampling for low-frequency signals
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Continuous SamplingContinuous Sampling
• Sampling multiplexed channels at constant rate.
• Causes phase skew btw. Channels– Use only if time relation btw. Channels is not
important
• Sampling multiplexed channels at constant rate.
• Causes phase skew btw. Channels– Use only if time relation btw. Channels is not
important
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Classic Multiplexed MIOClassic Multiplexed MIO
• Low cost/flexible– No anti-aliasing filters– Only one A/D converter for all channels
• Conflicts with some common requirements of many applications that require dynamic signal acquisition– Aliasing protection– Simultaneous sampling
• Low cost/flexible– No anti-aliasing filters– Only one A/D converter for all channels
• Conflicts with some common requirements of many applications that require dynamic signal acquisition– Aliasing protection– Simultaneous sampling
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Multiplexing: Some DefinitionsMultiplexing: Some Definitions
• Channels – the actual number of input channels scanned by the board
• Scan clock – the output data rate for each channel
• Decimation factor (D) – the acquisition over-sampling factor for each channel
• A/D clock – the actual sample rate of the multiplexing A/D converter
A/D clock = channels * decimation * scan clock
• Channels – the actual number of input channels scanned by the board
• Scan clock – the output data rate for each channel
• Decimation factor (D) – the acquisition over-sampling factor for each channel
• A/D clock – the actual sample rate of the multiplexing A/D converter
A/D clock = channels * decimation * scan clock
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Multiplexing Identical InputMultiplexing Identical Input
• 4 channels (same input signal on all channels)
• Scan clock = 1 kHz
• A/D clock = 4 kHz
• 4 channels (same input signal on all channels)
• Scan clock = 1 kHz
• A/D clock = 4 kHz
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Analog In
Chan. 0
Chan. 1
Chan. 2
Chan. 3
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Resulting Delayed AcquisitionsResulting Delayed Acquisitions
• Our four channels appear to have different phases even though we input the same signal to each
• Scan clock = 1 kHz• A/D clock = 4 kHz
• Our four channels appear to have different phases even though we input the same signal to each
• Scan clock = 1 kHz• A/D clock = 4 kHz
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Chan. 0
Chan. 1
Chan. 2
Chan. 3
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Relative Phase Responses: SkewRelative Phase Responses: Skew
• 4 channels
• Scan clock = 1 kHz
• A/D clock = 16 kHz (over-sampled 4X)
• 4 channels
• Scan clock = 1 kHz
• A/D clock = 16 kHz (over-sampled 4X)
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Chan. 0
Chan. 1
Chan. 2
Chan. 3
Hz
Degrees
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Additional Time Domain ConsiderationsAdditional Time Domain Considerations
• analog to digital converter– High resolution– Built-in anti-aliasing filters– Suited for sound and vibration measurements
• Simultaneous sampling and triggering– Phase relationship between signals
• Programmable gain
• Overload detection
• analog to digital converter– High resolution– Built-in anti-aliasing filters– Suited for sound and vibration measurements
• Simultaneous sampling and triggering– Phase relationship between signals
• Programmable gain
• Overload detection
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Time Domain ConsiderationsSmoothing WindowsTime Domain ConsiderationsSmoothing Windows
Nonintegral number of cyclesNonintegral number of cycles
• Reduces spectral leakage• Window selection depends on the application• PC Based instruments greatly facilitate transient
analysis
• Reduces spectral leakage• Window selection depends on the application• PC Based instruments greatly facilitate transient
analysis
No windowing
Windowing
Window
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Time vs Frequency DomainTime vs Frequency Domain
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Sample TimeDomain SignalSample Time
Domain Signal
FFTFFT
Anti-Alias Filter
Anti-Alias Filter
OctaveOctave
Acquire WaveformAcquire
Waveform
Basics of Frequency MeasurementsBasics of Frequency Measurements
SignalConditioning
SignalConditioning
FrequencyAnalysis
FrequencyAnalysis
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Frequency Domain AnalysisFrequency Domain Analysis
• FFT analysis
• Octave analysis
• Swept sine analysis
• FFT analysis
• Octave analysis
• Swept sine analysis
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FFT AnalysisFFT Analysis
• Time domain in discrete values Use Discrete Fourier Transform (DFT)
• Fast Fourier Transform (FFT)Optimized version of DFT
• Highest frequency that can be analyzed
• Frequency resolution
• Time domain in discrete values Use Discrete Fourier Transform (DFT)
• Fast Fourier Transform (FFT)Optimized version of DFT
• Highest frequency that can be analyzed
• Frequency resolution
2maxsfF fs : sampling frequencyfs : sampling frequency
N
f
Tf s
1 T : total acquisition time
N : FFT block size
T : total acquisition time
N : FFT block size
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FFT AnalysisFFT Analysis
• FFT gives magnitude and phase information– Magnitude = sqrt(Real^2 + Imag^2)– Phase = Tan-1(Imag / Real)
• Power Spectrum reflects the energy content– Power Spectrum = Mag^2
• Applications• Vibration analysis• Structural dynamics testing• Preventative maintenance• Shock testing
• FFT gives magnitude and phase information– Magnitude = sqrt(Real^2 + Imag^2)– Phase = Tan-1(Imag / Real)
• Power Spectrum reflects the energy content– Power Spectrum = Mag^2
• Applications• Vibration analysis• Structural dynamics testing• Preventative maintenance• Shock testing
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• Concentrates (“zooms”) FFT on a narrow band of frequencies
• Improves frequency resolution
• Distinguishes between closely-spaced frequencies
• Baseband analysis requires longer acquisition time for better resolution – requires more computation
• Concentrates (“zooms”) FFT on a narrow band of frequencies
• Improves frequency resolution
• Distinguishes between closely-spaced frequencies
• Baseband analysis requires longer acquisition time for better resolution – requires more computation
Zoom FFT AnalysisZoom FFT Analysis
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Zoom FFT AnalysisZoom FFT Analysis
Baseband FFT AnalysisBaseband FFT Analysis
Zoom FFTAnalysis
Zoom FFTAnalysis
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Octave AnalysisOctave Analysis
• Analysis performed through a parallel bank of bandpass filters– One octave corresponds to the doubling of the
frequency– Reference frequency is 1 kHz (audio domain)
• Analysis performed through a parallel bank of bandpass filters– One octave corresponds to the doubling of the
frequency– Reference frequency is 1 kHz (audio domain)
ff/2f/4 4 f2 f
1 octave
F
A
0220 Hz220 Hz 440 Hz440 Hz 880 Hz880 HzAA AAAA
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Octave AnalysisOctave Analysis
• Octave analysis gives log-spaced frequency information– Similar to human perception of sound
– 1/1, 1/3, 1/12, and 1/24 octave analysis
• FFT gives linearly-spaced frequency information
• Applications – noise emissions testing
– acoustic intensity measurement
– sound power measurement
– audio equalization
• Octave analysis gives log-spaced frequency information– Similar to human perception of sound
– 1/1, 1/3, 1/12, and 1/24 octave analysis
• FFT gives linearly-spaced frequency information
• Applications – noise emissions testing
– acoustic intensity measurement
– sound power measurement
– audio equalization
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Swept Sine AnalysisSwept Sine Analysis
• Source steps through a range of frequencies
• Analyzer measures frequency amplitude and phase at each step
• Non-FFT based
• Source steps through a range of frequencies
• Analyzer measures frequency amplitude and phase at each step
• Non-FFT based
Source
Device
Under Test
FrequencyResponse
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Auto-ranging: dynamic range optimized at each frequency• Adjust source amplitude• Adjust input range• Both improve dynamic range at particular frequencies
– Can get 140 dB effective dynamic range
Auto-ranging: dynamic range optimized at each frequency• Adjust source amplitude• Adjust input range• Both improve dynamic range at particular frequencies
– Can get 140 dB effective dynamic range
Swept Sine AnalysisSwept Sine Analysis
Gain
Chan A
Gain
Chan BSource Channel B
Channel A
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Swept Sine AnalysisSwept Sine Analysis
• Auto-resolution– Sweep optimized - more time at lower frequencies,
less time at higher– Increases frequency resolution on rapidly changing
responses
• Applications– Speaker testing– Cell phone testing– Electronic equipment characterization
• Auto-resolution– Sweep optimized - more time at lower frequencies,
less time at higher– Increases frequency resolution on rapidly changing
responses
• Applications– Speaker testing– Cell phone testing– Electronic equipment characterization
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Comparison of Frequency Analysis MethodsComparison of Frequency Analysis Methods
• FFT analysis– Very fast– Linear frequency scale– Based on discrete Fourier transform
• Octave analysis– Logarithmic frequency scale– Set of filters dividing frequency into bands– Similar to how human ear perceives sound
• Swept sine analysis– Good dynamic range– Source and analyzer step across frequency range – Slower response
• FFT analysis– Very fast– Linear frequency scale– Based on discrete Fourier transform
• Octave analysis– Logarithmic frequency scale– Set of filters dividing frequency into bands– Similar to how human ear perceives sound
• Swept sine analysis– Good dynamic range– Source and analyzer step across frequency range – Slower response
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Next LectureNext Lecture
• Output signals
• Servo-control systems
• Output signals
• Servo-control systems
