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Topic: Smoothing of data

Problem statement1. Produce a discrete time series by

super positioning 5 cosinusoidal components

for your own choice of the amplitudes and frequencies .



Problem statement2. Add some Gaussian noise to each

digitised value. Experiment with amplitudes (including that of the noise term), and frequencies, showing results

graphically. Then smooth your noisy time series with at least two different filters, e.g., a simple moving average smoother

and an order two binomial filter.



Problem statement3. Verify the validity of the

expression below, for your chosen smoother.


Part 1: Data Initialization

• The set of amplitudes chosen was [0.5, 1.0, 1.5, 2.0, 2.5].

• The set of frequencies chosen was [0.2, 04, 0.6, 0.8, 1.0].

• A floating point array of length 300 was created to represent the ‘time instants’ and the ‘noise’.

• A random number generator would then compound this noise to the original signal after multiplying by a factor of 0.5. Thus, a time-series was created, consisting of discrete amplitudes at different time instants.


Part 1: Generation of the sinusoidal plot

• A noiseless sinusoidal plot of the time series was generated by using the equation

which superimposed four different amplitudes

and four corresponding frequencies respectively.

• In the program, this was done by using two ‘for’ loops on the equation.


Part 1: Plots and Results


Plot of the sinusoidal signal created by superimposition of amplitudes and frequencies.

Part 2: Data Smoothing


The signal, with noise added with varying parameters,was smoothed using two different filters: 1. 5 point centred moving average filter.2. Binomial filter.

Part 2: 5 point centred moving average filter

The 5 point centred moving average filter was created using the following formula:

The use of this filter was found to remove the noise of the sinusoidal signal significantly by use of a 5-point moving average.


5 point centred moving average filter: Results


Plot of the smoothed signal, using a 5 point centred moving average filter

5 point centred moving average filter: Results


Plot of the smoothed signal vs that of the original noisy signal, using a 5 point centred moving average filter.

Part 2: Binomial Filter


• The binomial filter was implemented using the following formula.

• In this formula, we differ from the earlier in the aspect that the coefficients of the weighted terms are binomial coefficients of degree 5. The weight, 16, is the sum of all the individual weights.

Binomial Filter: Results


• Plot of the sinusoidal filter, smoothed using a 5th order binomial filter.

Part 3: Comparison of smoothing methods


The final part of the experiment was the comparison of smoothing done by each filtering method. For this, the mean and variance of the noisy signal and the smoothed signal, done by both filters, was compared.

Mean Mean difference Variance Variance difference

Original 0.012425 0 11.8078 0

5 point centred moving average filter

0.013343 0.0009183 11.271 0.5368

5th order binomial filter

0.012716 0.0002907 11.3401 0.4677

Results and Conclusions


1. The use of the 5 point moving average filter was found to remove the noise of the sinusoidal signal significantly by use of a 5-point moving average.

2. On the contrary, the binomial filter gives more weight to the point being replaced, since its coefficient is the highest among the five points being considered.

3. Since the mean and variance of the 5th order binomial filter were closer to the expected values, it is found to be the better of the two for smoothing noisy sinusoidal signals.

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