# H64DSP Digital Signal Processing

# Signal Enhancement

- Filtering in the frequency domain
- Multiplication-convolution feature of the Fourier transform and its applications
- Wiener filtering
- 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
*t**0* - The response to the signal lasts twice the signal
- Using in Baker Coding, CDMA, GPS

- the shape of the signal is

# 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)**

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

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

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 : \( A \cos(2\pi ft + \phi) \)

- generate 50 Hz signal then subtract from the signal
- 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