taking note

H64DSP Digital Signal Processing

H64DSP Digital Signal Processing


Signal Enhancement

  • Filtering in the frequency domain
  • Multiplication-convolution feature of the Fourier transform and its applications
  • Wiener filtering
    • weiner filter
      weiner filter
  • Averaging
    • provide signal improvement by 10 Log N
  • Matched filtering
    • the shape of the signal is known
    • Provides a highest possible spike at the output at t0
    • The response to the signal lasts twice the signal
    • Using in Baker Coding, CDMA, GPS

A/D and D/A conversion

  • Sampling of continuous signals
  • ADC parameters
  • ADC technologies
    • Ramp
    • Successive approximation
    • Flash
    • Pipeline
    • Sigma-Delta
  • Aliasing
    • Extra frequency components due to sampling – ALIASING
  • Complete digitiser
  • R-2R DAC and signal restoration
  • Fourier transform of discrete signals (DFT)

ADC errors

Quantization noise represents the difference between the input values to be digitised and the digitised ones. It can be reduced by increasing the effective number of bits of the ADC.

ADC saturation leads to inability to restore some input signal samples if their values are outside of the ADC range. Signal conditioning helps to match the ADC range with that of the output signal preventing the ADC saturation.

Insufficient sampling frequency leads to substantial part of the signal spectrum being aliased, i.e. appearing at frequencies that are different from those in the original signal. Increasing sampling frequency helps reducing the influence of aliasing.

Sampling clock jitter leads to sampling values that are not spaced exactly at the quantization interval from each other in the time domain, distorting the conversion results. This jitter can be reduced by using higher quality oscillator.


Digital Spectral analysis

Fast Fourier transform (FFT)

 

butterfly operation
butterfly operation

avoid complex multiplication

Zero-padding: increase FFT resolutions

Orthogonal frequency division multiplexing (OFDM)

Leakage of spectral components

Sharp discontinuities between original signal and periodic extensions.

These discontinuities introduce additional frequency components

Spectral leakage leads to appearance of weak spectral components, absent from the input signal, in the vicinity of a stronger spectral component that is present in the signal.

It appears due to the limited length of data record that is used for spectral analysis.

Leakage (a) masks weaker components that are actually present in the spectrum of the input signal and (b) make closely located spectral components look as a single component.

Windowing techniques

  • Discontinuities in extended periodic signal disappear, hence leakage is reduced
  • Multiplying by a window function distorts the signal, causing a broadening of the spectral peaks worsening resolution

Joint time-frequency analysis

Welch PSD

Both Welch PSD estimation and JTFA take sections of the process of interest that may overlap; both apply windows to these sections; both then calculate the Fourier transforms of these.
Welch procedure than squares the obtained spectra and calculates their average, presenting 2D graphs.

JTFA displays the spectra versus time as a 3D graph.

PSD estimation is useful for analysis of wide-sense stationary processes that retain their PSD fairly constant.
JTFA is useful for analysing time-varying (non-stationary) signals.

DFT vs FFT

The FFT is a computationally efficient algorithm to calculate the DFT for the equidistant grid of frequencies up to fs/2. It calculates N spectrum samples from the N input time domain samples.

The FFT requires Nlog2N operations whilst the direct DFT requires N2 operations for the same conditions. The higher efficiency of the FFT is based on the periodicity of the twiddle factors and use of the butterfly operation that implicitly does complex multiplication by basic twiddle factors.


Finite Impulse Response (FIR) digital Filter

output only depends on the weighted sum of input

no feedback, only feedforward

FIR filter can be made linear phase (all frequency delay by the same amount), achieved by symmetry of the coeff.

The conventional approach to FIR filter design is to minimise the difference between the required and obtained response by numerical optimisation

Hilbert Transform

linear phase + 90 degree phase shift

Hilbert Transform
Hilbert Transform

Used in:

  • envelope detection
  • integrator
  • differentiator

z-plane

Minimum Phase Filter

If all the zeroes are located inside the unity circle, the filter provides minimum phase response

Conversion:  for all zeros outside the unit circle, change a to 1/a

|H_old| = a|H_new|


Infinite Impulse Response (IIR) digital Filters

all pole should locate inside the unit circle

pole introduce gain as it get closer to the unit circle

Bilinear  transformation

squash the analog filter response to fit in fs/2, to avoid aliasing

There is change in -3dB frequency to compensate. (formula given in exam)

if fs >> fa(max) , fa can be assumed to be equal to fd

FIR vs. IIR

Just test table
FIR IIR
123 123
423 124

 

 


Adaptive Signal Processing

change h[m] coefficient according to the signal

Adaptive FIR filter

detect the feedback in the input signal and change the h[m] coefficient accordingly.

  • Not linear filter
  • not Linear Time Invariant (LTI) Signal
  • IIR adaptive filter can goes unstable as the poles exit the unit circle.
  • applicable to dynamically changing signal.
  • require reference. (model, measured. predicted)

Example: 50 Hz main notch filter also filter out the signal components at 50 Hz. (e.g. ECG signal) + notch has specified filter width, and fixed. Main frequency can change also.

  • Subtraction
    • generate 50 Hz signal then subtract from the signal
      • need to know the amplitude and phase : \( A \cos(2\pi ft + \phi) \)
  • Steepest descents
    • weighing, \( w^{m} \) coefficient is changed sample by sample, not block of data.
    • \(\mu\) : convergent perimeter: determine how convert of the algorithm
    • if the error is large, then the change weighing is large.
    • change in weight is proportional to the error, so it converges.
  • Adaptive subtraction
    • find the main aptitude and phase
    • get reference noise signal, highly correlated but not equal.
    • find optimal weighting coefficient w[p]
      • start with random weighting
      • apply FIR recursively
      • such that the MSE is minimised
      • try to fit Main signals as much as possible.
  • d: signal with noise
  • Reference, x,  need to be know! has to be correlated to the signal

Choice of \(\mu\)

  • too small: can’t track change fast enough
  • too large: oscillation in the output, or even unstable.
  • given between 2 limits

Adaptive Identification

Try to fit FIR filter to the plant. Comparing output version of the system to match, get weighting coefficient.

x: input to plant, d: plant output, y: plant output

such that y approximate d as much as possible.

Adaptive prediction

creating perdition model based on given previous datasets, trying to predict the x[n +1]

find model, \( w^{m} \),how x[m] depends previous x[m-L]; fit x[n] to the linear filter.  y: prediction of the time m. Then use the the same model to predict actual x[n+1].

note: x[p=0], present value, not including in the summation


Adaptive Signal Processing in Telecommunication

Adaptive equalisation

  • Transmission sequence of +1 and -1
  • dispersion effect in the channel \( H_{tc}(z) \) ,
  • We try to model the transfer function of the channel, by using inversion
  • However, the channel changes with time. Therefore the model need to be adaptive

Solution:

  • Transmit the training sequence, d(n), at the beginning of the transmitting message.
  • the training sequence is known at the receiver
  • adaptive calculate the weight of the model, such that the error is minimised.
  • Threshold detector is used to check convergence, if the filter is working correctly.
  • Model transfer function, H(z) approximate \( H_{tc}(z) \)
  • Decision Directed Equalisation: Using output from the threshold detector as the desired signal

Antenna

transmission mode: electrical signal -> electromagnetic signal

reception mode: electromagnetic signal -> electrical signal

Beam pattern: amplitude as the function of angle

  • Directivity is desired to get better signal
  • Array of dipole can give directivity
  • Dipole has no preferred direction of transmission (all direction)
  • 2 dipoles with in-phase signal, signals interfere constructively.
  • At certain angle, the signal can completely cancel each other. Referred as Null in the beam pattern.
  • MainLobe: preferred direction of transmission
  • SideLobe: unwanted direction of transmission
  • Null: no transmission at the angle.

Intensity of 2 Dipole

  • B is transmitting with \(\cos(2\pi f_0 t) \)
  • contribution form A is the delayed version of B, by t’
  • \(t’ = \frac{d \ sin (\theta)}{c}\)
  • A is transmitting with \(\cos(2\pi f_0 (t – t’)) \)
  • or, \(\cos(2\pi f_0 (t -\frac{d \ sin (\theta)}{c})) \)

MainLobe Width

Beam pattern related to Fourier Transform

???

Electronic Beam Steering

  • change the phase of the current
  • to compensate the phase shift of the array in that angle
  • subtract phase shift \(\Delta\phi\) to the current in beam B
  • get maximum transmission in the direction
  • normal direction is no longer in phase
  • can be done with computer control.

reduce side-lobe

  • apply windowing,with weighting coefficient.
  • increase the size of the main-lobe
  • thus, reduce resolution

 


Image Presentation and Processing

Colour image: 3D array, X and Y – pixel location, Z- RGB value

H in 2D array,  FIR in 2 dimentional

RGB2Gray for easier processing

Median Filter (non-linear filter) -get rid of the salt&pepper noise

Edge detection:

  • double differentiate,
  • edge where signal cross 0
  • need thresholding to avoid noise
  • differentiation is equivalent to high pass filter
  • more differentiation, more high frequency noise
  • need to lowpass signal then high-pass (essentially band pass)

gaussian filter

  • bell-shaped filter
  • narrower impulse response than moving-average filter
  • broader frequency domain

2D edge detection:

  • multiply 2 gaussian gaussian
  • second derivative -> laplacian
  • detect edge where laplacian changes sign

Image Compression

  • Lossless compression
    • looking for redundancy and compress
    • reversible method
    • winZip
    • Watermarking, astronomy research
  • Lossy compression
    • transform data and only keep the significant part
    • not reversible
    • JPEG
    • keep low frequency components
    • discard high frequency

Discrete Cosine Transform

  • special case of fouler transform
  • give real output

JPEG

  • Take DCT and keep the value
  • threshold the image and discard smaller values
  • transmit only coefficient within p-max and q-max
  • pad the rest with 0;
  • inverse DCT

 


Artificial Neural Network (ANN) and DSP implementation

calculates a weighted sum of the signals (stimuli) coming from different sources

ANNs usually consist of three layers; input layer is connected, e.g., to the intensity of pixels at the image that requires classification; hidden layer converts the outputs of the input layer neurons into the form suitable for the neurons in the output layer

At the output of the neuron there is a nonlinear element that produces the output signal. Most common non-linear function is sigmoid one