Week 1 - Basics of Estimation, Maximum Likelihood (ML) - Nptel

(1)

26/07/2020 Estimation for Wireless Communications – MIMO/OFDM Cellular and Sensor Networks - - Unit 2 - Week 1 - Basics o…

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Unit 2 - Week 1 - Basics

of Estimation, Maximum Likelihood (ML)

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Week 1 - Basics of Estimation, Maximum Likelihood (ML)

Lecture 01 - Basics – Sensor Network and Noisy Observation Model Lecture 02 - Likelihood Function and Maximum Likelihood (ML) Estimate Lecture 03 - Properties of Maximum Likelihood (ML) Estimate – Mean and Unbiasedness Lecture 04 - Properties of Maximum Likelihood (ML) Estimate – Variance and Spread Around Mean

Lecture 05 - Reliability of the Maximum Likelihood (ML) Estimate – Number of Samples Required Quiz : Assignment-1 Assignment-1 Solution

Due on 2017-08-06, 23:59 IST.

1 point 1)

1 point 2)

1 point 3)

Assignment-1

The due date for submitting this assignment has passed.

As per our records you have not submitted this assignment.

In the context of estimation, the probability density function (PDF) of the observations, viewed as a function of the unknown parameter is termed as the

Likelihood Function Estimation Function Objective Function Cost Function

No, the answer is incorrect.

Score: 0

Accepted Answers:

Likelihood Function

Consider the wireless sensor network (WSN) estimation scenario described in lectures with each observation , for , i.e. number of observations is . The ML estimate given by the sample mean has the following property.

It is biased

Gaussian distributed

Variance decreases as , where is number of observations All of the above

No, the answer is incorrect.

Score: 0

Gaussian distributed

Consider the wireless sensor network (WSN) estimation scenario described in lectures with each observation , for , i.e. number of observations is and noise samples are IID Gaussian noise samples with zero mean and variance . For this scenario, what is the maximum likelihood estimate of the unknown parameter .

ℎ

𝑦(𝑘) = ℎ + 𝑣(𝑘) 1 ≤ 𝑘 ≤ 𝑁 𝑁

1

𝑁²

𝑁

𝑦(𝑘) = ℎ + 𝑣(𝑘) 1 ≤ 𝑘 ≤ 𝑁 𝑁

𝑣(𝑘) 𝜎

²

ℎˆ ℎ

1

(𝑘)

𝑁

∑

𝑘=1 𝑁

𝑦

²

( ∏ 𝑦(𝑘))

𝑘=1

𝑁 𝑁¹

(2)

https://onlinecourses-archive.nptel.ac.in/noc17_ee19/unit?unit=86&assessment=93 2/4 Week 2 - Vector

Estimation

Week 3 - Cramer- Rao Bound (CRB), Vector Parameter Estimation, Multi-Antenna Downlink Mobile Channel

Estimation

Week 4 - Least Squares (LS) Principle, Pseudo-Inverse, Properties of LS Estimate, Examples – Mullti-Antenna Downlink and MIMO Channel Estimation

Week 5 - Inter Symbol Interference, Channel Equalization, Zero-forcing equalizer, Approximation error of equalizer

Week 6 - Introduction to Orthogonal Frequency Division Multiplexing (OFDM) and Pilot Based OFDM Channel Estimation, Example

Week 7 - OFDM – Comb Type Pilot (CTP)

Transmission, Channel Estimation in Time/ Frequency Domain, CTP Example, Frequency Domain Equalization (FDE), Example- FDE

Week 8 -

Sequential Least Squares (SLS) Estimation – Scalar/ Vector Cases, Applications - Wireless Fading Channel Estimation, SLS Example

1 point 4)

1 point 5)

1 point 6)

None of the above No, the answer is incorrect.

Score: 0

None of the above

Consider the wireless sensor network (WSN) estimation scenario described in lectures with each observation , for , i.e. number of observations is and noise samples are IID Gaussian noise samples with zero mean and variance . For this scenario, what is the variance of the maximum likelihood estimate of the unknown parameter

Score: 0

Consider the wireless sensor network (WSN) estimation scenario described in lectures with each observation , for , i.e. number of observations is and noise samples are IID Gaussian noise samples with zero mean and variance . For this scenario, what is the mean of the maximum likelihood estimate of the unknown parameter

Score: 0

Accepted Answers:

Consider now a slightly modified version of the wireless sensor network (WSN) estimation scenario described in class with each observation , for . Let the noise samples be IID Gaussian with mean and variance each. What is the maximum likelihood estimate

?

𝑚𝑖𝑛 {𝑦(𝑘), 1 ≤ 𝑘 ≤ 𝑁}

𝑦(𝑘) = ℎ + 𝑣(𝑘) 1 ≤ 𝑘 ≤ 𝑁 𝑁

𝑣(𝑘) 𝜎

²

ℎˆ ℎ

𝜎

²

𝜎² 𝑁 𝑁𝜎

𝑁𝜎

²

𝜎² 𝑁

𝑦(𝑘) = ℎ + 𝑣(𝑘) 1 ≤ 𝑘 ≤ 𝑁 𝑁

𝑣(𝑘) 𝜎

²

ℎˆ ℎ

ℎ 𝑁ℎ

𝑁ℎ

𝑁1

ℎ

𝑦(𝑘) = ℎ + 𝑣(𝑘) 1 ≤ 𝑘 ≤ 𝑁

𝜃 𝜎

²

ℎˆ

(𝑦(𝑘) − 𝜃

𝑁1

∑

𝑘=1

𝑁

)

²

(𝑦(𝑘) − 𝜃)

𝑁1

∑

𝑘=1 𝑁

(𝑦(𝑘) + 𝜃)

𝑁1

∑

𝑘=1 𝑁

(3)

https://onlinecourses-archive.nptel.ac.in/noc17_ee19/unit?unit=86&assessment=93 3/4 1 point 7)

1 point 8)

1 point 9)

No, the answer is incorrect.

Score: 0

Consider now a slightly modified version of the wireless sensor network (WSN) estimation scenario described in class with each observation , for . Let the noise samples be IID Gaussian with mean and variance each. What is mean of the maximum likelihood estimate ?

Score: 0

Consider now a slightly modified version of the wireless sensor network (WSN) estimation scenario described in class with each observation , for . Let the noise samples be lID Gaussian with mean and variance each. What is variance of the maximum likelihood estimate ?

Score: 0

Accepted Answers:

Consider now a slightly modified version of the wireless sensor network (WSN) estimation scenario described in class with each observation , for . Let the noise samples be IID Gaussian with mean and variance each. What is the distribution of the maximum likelihood estimate ?

Uniform Exponential Rayleigh

None of the above No, the answer is incorrect.

Score: 0 1

𝑦(𝑘)

𝑁

∑

𝑘=1 𝑁

(𝑦(𝑘) − 𝜃)

𝑁1

∑

𝑘=1 𝑁

𝑦(𝑘) = ℎ + 𝑣(𝑘) 1 ≤ 𝑘 ≤ 𝑁

𝜃 𝜎

²

ℎˆ

ℎ +

_𝑁^𝜃

ℎ

+ 𝜃

𝑁

ℎ + 𝜃 ℎ

ℎ

𝑦(𝑘) = ℎ + 𝑣(𝑘) 1 ≤ 𝑘 ≤ 𝑁

𝜃 𝜎

²

ℎˆ

+ 𝜎

²

𝜃

²

𝜎²

+

𝑁

𝜃

²

𝜎² 𝑁

𝜎²

+ 𝜃

𝑁

𝜎² 𝑁

𝑦(𝑘) = ℎ + 𝑣(𝑘) 1 ≤ 𝑘 ≤ 𝑁

𝜃 𝜎

²

ℎˆ

(4)

https://onlinecourses-archive.nptel.ac.in/noc17_ee19/unit?unit=86&assessment=93 4/4 1 point 10)

Accepted Answers:

None of the above

Consider now a slightly modified version of the wireless sensor network (WSN) estimation scenario described in class with each observation , for . Let the noise samples be IID Gaussian with mean and variance i.e. . What is the number of observations required such that the probability that the maximum likelihood estimate lies within a radius of of the unknown parameter is greater than ? Let Q denote the Gaussian Q function introduced in the lectures.

Score: 0

Accepted Answers:

𝑦(𝑘) = ℎ + 𝑣(𝑘) 1 ≤ 𝑘 ≤ 𝑁 𝜃 𝜎

²

= −6𝑑𝐵 10 log

₁₀

𝜎

²

= −6

𝑁 ℎˆ

1

16

ℎ 99.9%

(2 √ 𝑄 2‾

⁻¹

(𝜃 + 10

⁻³

) )

²

(8 𝑄

⁻¹

(5 × 10

⁻⁴

) )

²

8 𝑄

⁻¹

(5 × 10

⁻⁴

) 𝜃 + 2 𝑄

⁻¹

(5 √ 2‾ × 10

⁻⁴

)

(8 𝑄

⁻¹

(5 × 10

⁻⁴

) )

²

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