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Introduction to Signal Processing and Sampling and Reconstructing Continuous-time Signals Chaiwoot Boonyasiriwat August 14, 2020 What is Signal? Signal is a variable or function that contains information of a physical system.

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Chaiwoot Boonyasiriwat

August 14, 2020

Introduction to Signal Processing and Sampling and Reconstructing Continuous-time Signals

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▪ Signal is a variable or function that contains information of a physical system. ▪ Examples:

  • Audio and speech signals
  • Image and video signals
  • Medical signals: EEG, EKG, ultrasound
  • Geophysical signals: earthquakes, tide

gauge, LIDAR

  • Climate signals
  • SONAR, RADAR

What is Signal?

http://www.intechopen.com/source/html/18890/media/image4.jpeg

EEG Signal

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▪ Analog signal

  • Continuous signal with infinite resolution
  • Typically referred to electrical signal

converted from a physical variable by a transducer, e.g. microphone converting audio signal into electrical (analog) signal. ▪ Discrete signal: signal values are available at some discrete points in space or time ▪ Digital signal: value of discrete signal is stored with a finite precision, e.g., in computer as fixed-point or floating-point numbers.

Types of Signals

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A discrete-time signal can be obtained by taking samples of an analog signal where T is the sampling interval or time between samples, and sampling frequency or sampling rate (Hz). “When finite precision is used to represent the value of , the sequence of quantized values is called a digital signal.”

Sampling

Schilling and Harris (2012; p.3,12)

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In exploration seismology, signals are in digital form. Any system or algorithm which processes input digital signal and produces an output digital signal is a digital signal processor.

Digital Signal Processor

Schilling and Harris (2012, p.3)

S

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Causal signal: Acausal signal: Examples of causal signals: Heaviside step function: Dirac delta function:

Causal and Acausal Signal

Schilling and Harris (2012, p.15-16)

Sifting property of delta function

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▪ Continuous system: continuous input and output

▪ Discrete system: discrete input and output ▪ Linear system: ▪ Time-invariant system: ▪ Bounded signal: ▪ Stable system: every bounded input produces a bounded output (BIBO)

System Classification

Schilling and Harris (2012, p.16-18)

S

input system output

System is considered as a function or operator:

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▪ Forward and inverse Fourier transforms: ▪ Magnitude spectrum: ▪ Phase spectrum: ▪ Polar form: ▪ “For real , the magnitude spectrum is even function of f, and the phase spectrum is odd function of f.”

Magnitude and Phase

Schilling and Harris (2012, p.18-19)

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▪ Filter is a system designed to reshape the spectrum of a signal. ▪ For linear time-invariant (LTI) continuous-time system S with input and output , the frequency response is defined as where is magnitude response of S is phase response of S

Filter and Frequency Response

Schilling and Harris (2012, p.19)

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▪ Impulse response is the system output when the input is the unit impulse (Dirac function) ▪ It can be shown that ▪ As a result, when the system input is the unit impulse, the frequency response of the system is

Impulse Response

Schilling and Harris (2012, p.20)

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Ideal low-pass filter with cut-off frequency B has frequency response Recall that So, the frequency component of in the range [- B, B] passes through the filter without distortion.

Example of Continuous-Time System

Schilling and Harris (2012, p.20)

( )

a

H f

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Using the inverse Fourier transform on the frequency response, we obtain the impulse response of this filter as where

Impulse Response of Ideal Low-pass Filter

Schilling and Harris (2012, p.20-21)

Sinc function is an acausal output of the filter when the input is causal. “So, this filter cannot be realized with physical hardware.”

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Periodic impulse train with period T is defined as The sampled version of signal denoted by is defined as

Amplitude modulation

Sampling of Continuous Signals

Schilling and Harris (2012, p.21-23)

( ) 10

t a

x t te− =

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We then have

Sampling as Modulation

Schilling and Harris (2012, p.22)

Using the property

x(n) is a discrete-time signal.

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Note that is still a continuous-time signal. If is causal, we can apply the Laplace transform to the signal. For causal signals, the Fourier transform is the Laplace transform with Therefore, the spectrum of a causal signal can be obtained from its Laplace transform, i.e.,

Laplace Transform

Schilling and Harris (2012, p.22-23)

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Taking the Laplace transform of and then using , we then obtain the spectrum of , the sampled version of , as where

Spectrum of Sampled Signal

Schilling and Harris (2012, p.23)

Aliasing formula

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“A continuous-time signal is band-limited to bandwidth B if and only if its magnitude spectrum satisfies .”

Band-Limited Signal

https://upload.wikimedia.org/wikipedia/commons/thumb/f/f7/ Bandlimited.svg/300px-Bandlimited.svg.png

Schilling and Harris (2012, p.23)

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“The spectrum of the sampled version of a signal is a sum of scaled and shifted spectra of the original signal.” A replicated version of the spectrum of original signal is scaled by fs, the sampling frequency , and shifted by nfs where n is integer.

Aliasing Formula

Schilling and Harris (2012, p.23)

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Aliasing Formula

http://archives.sensorsmag.com/articles/0103/38/fig5_big.gif

  • fs
  • 2fs
  • 2fs
  • fs

Spectrum of the original signal Scaled and shifted replicas of the original spectrum

Base band Side bands Side bands

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The overlap of the replicas of the original spectrum (aliasing) occurs when .

Aliasing

http://archives.sensorsmag.com/articles/0103/38/fig5_big.gif

  • fs
  • 2fs
  • 2fs
  • fs

The original signal can no longer be exactly reconstructed! Why?

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Aliasing

Schilling and Harris (2012, p.24)

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“Suppose a continuous-time signal is band- limited to B Hz. Let denote the sampled version of using impulse sampling with a sampling frequency . Then the samples contain all the information necessary to recover the original signal if .”

Shannon Sampling Theorem

http://zone.ni.com/images/reference/ en-XX/help/370051T-01/aliasing_effects.gif

Schilling and Harris (2012, p.25)

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Nyquist frequency is defined as If has any frequency component outside of then in these frequencies get reflected about and folded back into the range This is call aliasing.

Nyquist Frequency

Schilling and Harris (2012, p.26)

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Undersampling: ▪ Aliasing occurs. ▪ Original signal cannot be reconstructed. Oversampling: ▪ No aliasing. ▪ Original signal can be reconstructed.

Under- and Over-Sampling

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Consider the signal The spectrum of is

Example: Aliasing

Adapted from Example 1.5 of Schilling and Harris (2012, p.25)

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To avoid aliasing, we need Hz. Let’s sample at the rate Hz. In this case, s and the samples are

Example: Aliasing

Samples of are identical to those of

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Example: Aliasing

O

( ) and ( )

a

x t x n ( )

a

X f ˆ ( )

a

X f ˆ ( ) X f 75

n

f = 150

s

f =

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▪ Signal reconstruction is to recover the spectrum from the spectrum . ▪ This can be done by passing through an ideal lowpass reconstruction filter that removes the side bands and rescales the base band.

Schilling and Harris (2012, p.27)

Reconstruction of Continuous Signal

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Reconstruction of Continuous Signal

Remove side bands Remove side bands

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The impulse response of the ideal reconstruction filter is The original continuous signal can be perfectly reconstructed by

Schilling and Harris (2012, p.28)

Reconstruction of Continuous Signal

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Fourier transform of the convolution of a(t) and b(t), denoted as a(t)*b(t), is the pointwise product of their Fourier transforms: where It can be shown that convolution is commutative, that is

Convolution Theorem

https://en.wikipedia.org/wiki/Convolution_theorem

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Schilling and Harris (2012, p.28)

Shannon Interpolation Formula

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“Suppose a continuous-time signal is bandlimited to B Hz. Let be the n-th sample of using a sampling frequency

  • f . If , then can be

reconstructed from as follows.” The sinc function is used here as a basis function for interpolation.

Schilling and Harris (2012, p.28)

Shannon Interpolation Formula

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“Let be a causal nonzero input to a continuous-time linear system, and let be the corresponding output. The transfer function

  • f the system is defined as”

Since the transfer function is the Laplace transform of the impulse response

Schilling and Harris (2012, p.29)

Transfer Function

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Consider the system . So,

Schilling and Harris (2012, p.29-30)

Example: Time Shift

) exp( i −

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▪ Exact reconstruction of using the Shannon interpolation formula required an ideal filter which cannot be realized by a physical system. ▪ The reconstruction can be approximated by a practical filter such as a zero-order hold filter ▪ The impulse response of a zero-order hold is

Zero-Order Hold

Schilling and Harris (2012, p.30) / t T

0( )

h t

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▪ The zero-order hold is linear and time-invariant. ▪ The response to an impulse of strength at time will be a pulse of height and width starting at ▪ When the input is , the output will be a piecewise-constant approximation to ▪ The transfer function of zero-order hold is

Zero-Order Hold

Schilling and Harris (2012, p.31)

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Zero-order hold can be used as a digital-to-analog converter (DAC) while an impulse sampler can be used as an analog-to-digital converter (ADC).

Digital Signal Processing System

Schilling and Harris (2012, p.31-32) Switch opens and closes every T seconds.

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▪ When a signal that is not bandlimited is sampled, aliasing will occur. To avoid aliasing, a lowpass filter must be applied to the signal. ▪ An anti-aliasing filter is a lowpass filter that removes all frequency components outside range where is called the cut-off frequency. ▪ The ideal lowpass filter is the optimal choice for an anti-aliasing filter. Butterworth filter is a practical filter that has been widely used as an anti-aliasing filter.

Anti-aliasing Filter

Schilling and Harris (2012, p.33)

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▪ “A lowpass Butterworth filter of order n has the magnitude response as follows.” ▪ At the cut-off frequency fc, and so fc is called the 3 dB cutoff frequency of the filter.

Butterworth Filter

Schilling and Harris (2012, p.33)

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The transfer function of a lowpass Butterworth filter of order n is

Butterworth Filter

Schilling and Harris (2012, p.33-34)

“As the order n increases, the magnitude response approaches the ideal lowpass characteristic.”

1

c

 =

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▪ The transfer function of the first-order Butterworth filter is ▪ The circuit realization of the first-order Butterworth filter is shown below. ▪ This circuit requires

3 operational amplifiers 6 resistors of resistance R capacitor of capacitance C and

First-Order Butterworth Filter

Schilling and Harris (2012, p.34-35)

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▪ Schilling, R. J. and S. L. Harris, 2012, Fundamentals of Digital Signal Processing using MATLAB, Second Edition, Cengage Learning.

References