# Open-book take-home examination

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*Open-book take-home examination *

# Answer ALL questions

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- (a) It is required to reduce the effects of noise and interference on a signal with samples {
*x*[*n*]}, using the DFT method.

There are 1024 samples and a sampling frequency of 512 Hz is used.

The signal has significant frequency components between 100 Hz and 200 Hz. White noise is present at all frequencies. There is also an interference at 150 Hz.

Explain how you would apply DFT methods to reduce the effects of noise

and interference from the signal giving details as to how you process the

DFT and how you obtain the final filtered signal. [15]

- It is now required to apply the Wiener filter to {
*x*[*n*]} in part (a). Explain the processing steps that are required to obtain the final filtered signal. [10]

- (a) Averaging is used to enhance above noise the estimate of one occurrence of a repetitive signal in noise.

- Explain, giving an illustrative example, what is meant by
*coherent*

averaging*.* [3]

- Explain why incoherent averaging (i.e. averaging that is not coherent) occurs if the signal to noise ratio is too small. [3]

- Explain the principle of matched filtering and how it can be used to

help achieve coherent averaging. [6]

- In Figure Q2(b), two electrocardiogram signals, taken from the expectant mother, are shown.

Figure Q2(b)

## Continued on the next page

The top signal is measured from the mother’s abdomen. M1 refers to the times of occurrence of the maternal contribution to the abdominal signal and F refers to the fetal contribution.

The bottom signal is measured from the mother’s chest and this signal consists of just the maternal ECG signal; the times of occurrence are marked as M2.

The maternal contributions, M1 and M2, are strongly correlated but not equal.

Explain how an adaptive filter can be applied to estimate the fetal ECG

signal (F). [13]

- (a) You are monitoring a medical signal that has dominant frequency components between 220 Hz and 280 Hz.

There is mains interference at 50 Hz and harmonics at 100 Hz and 150 Hz. There is background white noise at all frequencies.

Design an FIR linear phase filter to reduce the effects of noise and interference on the medical signal, indicating the following:

- Your choice of sampling frequency. [3]
- placement of zeros and poles in the z-plane. [6] iii) the transfer function,
*H*(*z*), for the designed filter. [6] iv) the relation between the output and input samples for the filter. [3]

(b) The analytical signal, *s _{a}*(

*t*), for

*s*(

*t*) is given by:

?_{?}(?) = ?(?) + ? ?̂(?)

Where ?̂(?) is the Hilbert transform of *s*(*t*).

Describe how you would use the analytical signal to track the instantaneous frequency of a signal consisting of __two__ sinusoids, with different frequencies:

?(?) = cos(2??_{1}?) + cos(2??_{2}?)

indicating any additional processing methods that need to be applied. [7]

- (a) The median filter is used to eliminate spikes from data. Sometimes,

when applied once, spikes still remain. Discuss the advantages and

disadvantages of applying the median filter further times to the data to

try to eliminate these residual spikes. [4]

- Starting with the signal and spectrum for the Discrete Time Fourier Series

(DTFS), and using the convolution theorem, derive the signal and

spectrum for the Discrete Time Fourier Transform (DTFT). [7]

- You are estimating a signal which is known to consist mainly of dc segments, with occasional linear ramps – an example is shown in Figure Q4(c). Noise is present throughout the signal, as indicated in this Figure.

Figure Q4(c)

Explain how you would estimate the complete signal above using a Kalman filter. Your explanation should include:

- The model or models that you would use for the signal variation

between samples. [4]

- How you would take into account the change of model from dc to ramp and from ramp to dc, with justification for your chosen

method. [10]

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