Least Squares Estimation in Signal Analysis
Least squares estimation is a method used to approximate solutions when exact solutions are not available.
Mathematical Formulation
Given a linear model
we minimize the squared error
The solution is
Interpretation
Least squares finds the best approximation by minimizing the total error energy. It finds the parameters that make the model fit the data as closely as possible in a mean-square sense.
Applications in Signal Analysis
1. Signal Fitting
Approximate a signal using a known model.
Suppose a signal is modeled as
Using least squares, we can estimate
- Amplitude A
- Frequency ω
- Phase ϕ
2. System Identification
Estimate system parameters from input-output data.
Suppose FIR filter is modeled as
Using least squares, we can estimate impulse response h(n).
3. Noise Reduction (Denoising)
Separate signal from noise. The signal can be expressed as a weighted sum of basis functions.
Measured signal
Suppose measured signal is modeled as
Using least squares, we can estimate coeffient (weight) ck where ϕk is basis function.
4. Trend Estimation
Extract slow-varying components.
Fit a polynomial
Using least squares, we can estimate fitting coefficients a0, a1, and a2 in this case.
Comparison of raw data, linear polynomial fit, and exponential fit results (refer to Samples/curve fitting.mmj)
5. Overdetermined systems
Solve systems with more equations than unknowns. Many measurements while a few unknown parameters.
Using least squares, we can estimate unknown parameters with redundant measurements.
Conclusion
Least squares finds the best-fit model by minimizing the total squared error between measured and predicted signals.
It enables
- Signal modeling
- System identification
- Noise reduction
- Parameter estimation
- Solving overdetermined systems
Suggested Further Reading
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Least Squares Estimation in Signal Analysis
Least squares estimation is a method used to approximate solutions when exact solutions are not available.
Mathematical Formulation
Given a linear model
we minimize the squared error
The solution is
Interpretation
Least squares finds the best approximation by minimizing the total error energy. It finds the parameters that make the model fit the data as closely as possible in a mean-square sense.
Applications in Signal Analysis
1. Signal Fitting
Approximate a signal using a known model.
Suppose a signal is modeled as
Using least squares, we can estimate
2. System Identification
Estimate system parameters from input-output data.
Suppose FIR filter is modeled as
Using least squares, we can estimate impulse response h(n).
3. Noise Reduction (Denoising)
Separate signal from noise. The signal can be expressed as a weighted sum of basis functions.
Measured signal
Suppose measured signal is modeled as
Using least squares, we can estimate coeffient (weight) ck where ϕk is basis function.
4. Trend Estimation
Extract slow-varying components.
Fit a polynomial
Using least squares, we can estimate fitting coefficients a0, a1, and a2 in this case.
Comparison of raw data, linear polynomial fit, and exponential fit results (refer to Samples/curve fitting.mmj)
5. Overdetermined systems
Solve systems with more equations than unknowns. Many measurements while a few unknown parameters.
Using least squares, we can estimate unknown parameters with redundant measurements.
Conclusion
Least squares finds the best-fit model by minimizing the total squared error between measured and predicted signals.
It enables
Suggested Further Reading
#You may also find these topics helpful: