In this project, I implemented a minimum mean squared error (MMSE) channel estimation of low complexity using techniques from the field of machine learning. The covariance matrices of these channels vectors depend on random parameters. For example, the random parameter in this project is the angles of propagation paths. If the covariance matrix exhibits certain Toeplitz and shift-invariance structures, the complexity of the MMSE channel estimator is in the order of 훩(푀 푙표푔푀) (푀 is the channel dimension or, in this project, the number of antennas at the base station) floating point operations, otherwise 훩(푀3). In this case, an efficient suboptimal estimator with low complexity is obtained by using the MMSE estimator of the structured model as a blueprint for the architecture of a neural network. This network learns the MMSE estimator for the unstructured model, but only within the given class of estimators, that contains the MMSE estimator for the structured model.

1. Learning the MMSE Channel Estimator
EESC 6353 Project
Shamman Noor

2. Introduction
• System Model
• MMSE Estimation:
- Discrete MMSE estimation
- Structured MMSE estimation
- Circulant structure
- Toeplitz structure
- Fast MMSE estimation
- CNN based fast MMSE estimation
• CNN Model and parameters
Software used:
• MATLAB – 2018a
• Python – 3.7
• TensorFlow – 1.13.1

3. System Model
…
Base station
(M antennas)
UE
(single antenna)
…
𝑦𝑡 = ℎ 𝑡 + 𝑧𝑡
ℎ1,1
ℎ1,2
ℎ1,𝑀 𝑧𝑡 ~ 𝑁𝐶(0, Σ)
Σ = 𝜎2 𝐼 ~ 𝑎𝑠𝑠𝑢𝑚𝑒𝑑
𝜎2 ~ 𝑘𝑛𝑜𝑤𝑛
ℎ 𝑡|𝛿 ~ 𝑁𝐶(0, 𝐶 𝛿)
𝛿 ~ 𝑝(𝛿)
𝑦𝑡|ℎ 𝑡 ~ 𝑁𝐶(ℎ 𝑡, Σ)
𝑇ℎ𝑒 𝑔𝑜𝑎𝑙 𝑖𝑠 𝑡𝑜 𝑒𝑠𝑡𝑖𝑚𝑎𝑡𝑒 𝒉 𝒕,
𝑓𝑟𝑜𝑚 𝑟𝑒𝑐𝑒𝑖𝑣𝑒𝑑 𝑠𝑖𝑔𝑛𝑎𝑙, 𝑦𝑡
𝑎𝑛𝑑 𝑡ℎ𝑒 𝑑𝑖𝑠𝑡𝑟𝑖𝑏𝑢𝑡𝑖𝑜𝑛𝑠 𝑜𝑓ℎ 𝑡, 𝑦𝑡 𝑎𝑛𝑑 𝑧𝑡

4. System Model
…
Base station
(M antennas)
UE
(single antenna)
…
ℎ1,1
ℎ1,2
ℎ1,𝑀

5. MMSE channel estimation
𝑊𝛿 = 𝐶 𝛿(𝐶 𝛿 + 𝛴)−1
𝐶 =
1
𝜎2
𝑡=1
𝑇
𝑦𝑡 𝑦𝑡
𝐻 𝑊∗ 𝐶 =
𝑒𝑥𝑝(𝑡𝑟 𝑊𝛿 𝐶 + 𝑇𝑙𝑜𝑔 𝐼 − 𝑊𝛿 ) 𝑊𝛿 𝑝(𝛿) 𝑑𝛿
𝑒𝑥𝑝(𝑡𝑟 𝑊𝛿 𝐶 + 𝑇𝑙𝑜𝑔 𝐼 − 𝑊𝛿 ) 𝑝(𝛿) 𝑑𝛿
𝐶 𝛿 = 𝑐𝑜𝑣(ℎ 𝑡)
𝑝(𝛿)Distribution of channel parameters,
𝑦𝑡
𝜎2Know noise variance,
Received signal,
Channel covariance matrix
Sample covariance matrix
(estimated)
MMSE filter coefficients
MMSE filter (estimated)
ℎ = 𝑊∗ 𝐶 𝑦
Estimated channel
ℎ 𝑡Generated channel,
General MMSE Estimator:
𝐸 ℎ 𝑡 𝑌, 𝛿 = 𝑊𝛿 𝑦𝑡
ℎ 𝑡 = 𝐸 ℎ 𝑡 𝑌 = 𝑊∗ 𝐶 𝑦𝑡

6. MMSE channel estimation
𝐶 =
1
𝜎2
𝑡=1
𝑇
𝑦𝑡 𝑦𝑡
𝐻 𝑊∗ 𝐶 =
𝑒𝑥𝑝(𝑡𝑟 𝑊𝛿 𝐶 + 𝑇𝑙𝑜𝑔 𝐼 − 𝑊𝛿 ) 𝑊𝛿 𝑝(𝛿) 𝑑𝛿
𝑒𝑥𝑝(𝑡𝑟 𝑊𝛿 𝐶 + 𝑇𝑙𝑜𝑔 𝐼 − 𝑊𝛿 ) 𝑝(𝛿) 𝑑𝛿
𝑝(𝛿)Distribution of channel parameters,
Sample covariance matrix
(estimated)
MMSE filter (estimated)
ℎ = 𝑾 𝜹 𝑦
Estimated channel
Genie Aided MMSE Estimator (for a lower bound performance):
𝑦𝑡
𝜎2Know noise variance,
Received signal,
𝑾 𝜹 = 𝐶 𝛿(𝐶 𝛿 + 𝛴)−1𝐶 𝛿 = 𝑐𝑜𝑣(ℎ 𝑡)
Channel covariance matrix MMSE filter coefficients
ℎ 𝑡Generated channel,

7. MMSE channel estimation
𝐶 =
1
𝜎2
𝑡=1
𝑇
𝑦𝑡 𝑦𝑡
𝐻 𝑾 𝑮𝑬 𝑪 =
𝑒𝑥𝑝(𝑡𝑟 𝑊𝛿 𝐶 + 𝑇𝑙𝑜𝑔 𝐼 − 𝑊𝛿 ) 𝑊𝛿 𝑝(𝛿) 𝑑𝛿
𝑒𝑥𝑝(𝑡𝑟 𝑊𝛿 𝐶 + 𝑇𝑙𝑜𝑔 𝐼 − 𝑊𝛿 ) 𝑝(𝛿) 𝑑𝛿𝑦𝑡
𝜎2Know noise variance,
Received signal,
Sample covariance matrix
(estimated)
MMSE filter (estimated)
ℎ = 𝑾 𝑮𝑬 𝑪 𝑦
Estimated channel
Gridded MMSE Estimator:
𝑊𝛿 = 𝐶 𝛿(𝐶 𝛿 + 𝛴)−1𝐶 𝛿 = 𝑐𝑜𝑣(ℎ 𝑡)
Channel covariance matrix MMSE filter coefficients
ℎ 𝑡Generated channel,
Assumption 1: 𝑝 𝛿𝑖 =
1
𝑁
, ∀𝑖 = 1, … , 𝑁
N = 16M
M = number of antennas

8. MMSE channel estimation
𝐶 =
1
𝜎2
𝑡=1
𝑇
𝑦𝑡 𝑦𝑡
𝐻 𝑾 𝑮𝑬 𝑪 =
𝑒𝑥𝑝(𝑡𝑟 𝑊𝛿 𝐶 + 𝑇𝑙𝑜𝑔 𝐼 − 𝑊𝛿 ) 𝑊𝛿 𝑝(𝛿) 𝑑𝛿
𝑒𝑥𝑝(𝑡𝑟 𝑊𝛿 𝐶 + 𝑇𝑙𝑜𝑔 𝐼 − 𝑊𝛿 ) 𝑝(𝛿) 𝑑𝛿
Assumption 1: 𝑝 𝛿𝑖 =
1
𝑁
, ∀𝑖 = 1, … , 𝑁
N = 16M
M = number of antennas
𝑦𝑡
𝜎2Know noise variance,
Received signal,
Sample covariance matrix
(estimated)
MMSE filter (estimated)
ℎ = 𝑾 𝑮𝑬 𝑪 𝑦
Estimated channel
Gridded MMSE Estimator :
𝑣𝑒𝑐 𝑊𝛿 ∈ 𝐂 𝑀2 𝑥𝑁
→ 𝛰(𝑀2
𝑁) → 𝑂(𝑀3
)
𝑊𝛿 = 𝐶 𝛿(𝐶 𝛿 + 𝛴)−1𝐶 𝛿 = 𝑐𝑜𝑣(ℎ 𝑡)
Channel covariance matrix MMSE filter coefficients
ℎ 𝑡Generated channel,

9. MMSE channel estimation
𝑪 =
𝟏
𝝈 𝟐
𝒕=𝟏
𝑻
𝑸𝒚 𝒕
𝟐 𝑾 𝑺𝑬 𝑪 =
𝑒𝑥𝑝(𝑡𝑟 𝑊𝑆𝐸 𝐶 + 𝑇𝑙𝑜𝑔 𝐼 − 𝑄 𝐻 𝑊𝛿 𝑄 ) 𝑊𝑆𝐸 𝑝(𝛿) 𝑑𝛿
𝑒𝑥𝑝(𝑡𝑟 𝑊𝑆𝐸 𝐶 + 𝑇𝑙𝑜𝑔 𝐼 − 𝑄 𝐻 𝑊𝛿 𝑄 ) 𝑝(𝛿) 𝑑𝛿
𝑝(𝛿)Distribution of channel parameters,
𝑦𝑡
𝜎2Know noise variance,
Received signal,
Sample covariance matrix
(estimated)
MMSE filter (estimated)
ℎ = 𝑄 𝐻 𝑾 𝑺𝑬 𝑪 𝑄𝑦
Estimated channel
Structured MMSE Estimator:
Assumption 2: 𝑊𝛿 = 𝑄 𝐻
𝑑𝑖𝑎𝑔 𝑤 𝛿 𝑄
𝑤 𝛿 = 𝑑𝑖𝑎𝑔(𝑄𝑊𝛿 𝑄 𝐻
)
𝑄 =
𝐹, 𝑓𝑜𝑟 𝑐𝑖𝑟𝑐𝑢𝑙𝑎𝑛𝑡 𝑒𝑠𝑡𝑖𝑚𝑎𝑡𝑜𝑟
𝐹2, 𝑓𝑜𝑟 𝑡𝑜𝑒𝑝𝑙𝑖𝑡𝑧 𝑒𝑠𝑡𝑖𝑚𝑎𝑡𝑜𝑟
𝑊𝑆𝐸 = 𝑄𝑊𝛿 𝑄 𝐻𝑊𝛿 = 𝐶 𝛿(𝐶 𝛿 + 𝛴)−1𝐶 𝛿 = 𝑐𝑜𝑣(ℎ 𝑡)
Channel covariance matrix MMSE filter coefficients
ℎ 𝑡Generated channel,

10. MMSE channel estimation
𝑪 =
𝟏
𝝈 𝟐
𝒕=𝟏
𝑻
𝑸𝒚 𝒕
𝟐 𝑾 𝑺𝑬 𝑪 =
𝑒𝑥𝑝(𝑡𝑟 𝑊𝑆𝐸 𝐶 + 𝑇𝑙𝑜𝑔 𝐼 − 𝑄 𝐻 𝑊𝛿 𝑄 ) 𝑊𝑆𝐸 𝑝(𝛿) 𝑑𝛿
𝑒𝑥𝑝(𝑡𝑟 𝑊𝑆𝐸 𝐶 + 𝑇𝑙𝑜𝑔 𝐼 − 𝑄 𝐻 𝑊𝛿 𝑄 ) 𝑝(𝛿) 𝑑𝛿
𝑝(𝛿)Distribution of channel parameters,
𝑦𝑡
𝜎2Know noise variance,
Received signal,
Sample covariance matrix
(estimated)
MMSE filter (estimated)
ℎ = 𝑄 𝐻 𝑾 𝑺𝑬 𝑪 𝑄𝑦
Estimated channel
Structured MMSE Estimator:
𝑊𝑆𝐸 = 𝑄𝑊𝛿 𝑄 𝐻
𝑊𝑆𝐸 ∈ 𝐂 𝑀𝑥𝑁
→ 𝛰(𝑀𝑁) → 𝑂(𝑀2
)
𝑊𝛿 = 𝐶 𝛿(𝐶 𝛿 + 𝛴)−1𝐶 𝛿 = 𝑐𝑜𝑣(ℎ 𝑡)
Channel covariance matrix MMSE filter coefficients
ℎ 𝑡Generated channel,
Assumption 2: 𝑊𝛿 = 𝑄 𝐻
𝑑𝑖𝑎𝑔 𝑤 𝛿 𝑄
𝑤 𝛿 = 𝑑𝑖𝑎𝑔(𝑄𝑊𝛿 𝑄 𝐻
)
𝑄 =
𝐹, 𝑓𝑜𝑟 𝑐𝑖𝑟𝑐𝑢𝑙𝑎𝑛𝑡 𝑒𝑠𝑡𝑖𝑚𝑎𝑡𝑜𝑟
𝐹2, 𝑓𝑜𝑟 𝑡𝑜𝑒𝑝𝑙𝑖𝑡𝑧 𝑒𝑠𝑡𝑖𝑚𝑎𝑡𝑜𝑟

11. MMSE channel estimation
𝑾 𝑭𝑬 𝑪 =
𝑒𝑥𝑝(𝑡𝑟 𝐹 𝐻 𝑾 𝟎 𝐹 𝐶 + 𝑇𝑙𝑜𝑔 𝐼 − 𝑾 𝑭𝑬 ) 𝐹 𝐻 𝑾 𝟎 𝐹𝑝(𝛿) 𝑑𝛿
𝑒𝑥𝑝(𝑡𝑟 𝐹 𝐻 𝑾 𝟎 𝐹 𝐶 + 𝑇𝑙𝑜𝑔 𝐼 − 𝑾 𝑭𝑬 ) 𝑝(𝛿) 𝑑𝛿
𝑝(𝛿)Distribution of channel parameters,
𝑦𝑡
𝜎2Know noise variance,
Received signal,
Sample covariance matrix
(estimated)
MMSE filter (estimated)
ℎ = 𝐹 𝐻 𝑾 𝑭𝑬 𝑪 𝐹𝑦
Estimated channel
Fast MMSE Estimator:
Generated channel, 𝑾 𝑭𝑬 = 𝐶𝑐(𝐶𝑐 + 𝛴)−1𝐶 𝛿 = 𝑐𝑜𝑣(ℎ 𝑡)
Channel covariance matrix MMSE filter coefficients
ℎ 𝑡Generated channel, 𝑪 𝒄 = 𝑐𝑖𝑟𝑐(𝐶 𝛿)
Circulant approximation of
channel covariance matrix
𝑾 𝟎 = 𝐹𝑾 𝑭𝑬
FFT of filter
𝑪 =
𝟏
𝝈 𝟐
𝒕=𝟏
𝑻
𝐹𝒚 𝒕
𝟐
Assumption 3: 𝐯𝐞𝐜(𝑊𝛿) = 𝐹 𝐻
𝑑𝑖𝑎𝑔 𝐹𝑤 𝛿 𝐹

12. MMSE channel estimation
𝑾 𝑭𝑬 𝑪 =
𝑒𝑥𝑝(𝑡𝑟 𝐹 𝐻 𝑾 𝟎 𝐹 𝐶 + 𝑇𝑙𝑜𝑔 𝐼 − 𝑾 𝑭𝑬 ) 𝐹 𝐻 𝑾 𝟎 𝐹𝑝(𝛿) 𝑑𝛿
𝑒𝑥𝑝(𝑡𝑟 𝐹 𝐻 𝑾 𝟎 𝐹 𝐶 + 𝑇𝑙𝑜𝑔 𝐼 − 𝑾 𝑭𝑬 ) 𝑝(𝛿) 𝑑𝛿
𝑝(𝛿)Distribution of channel parameters,
𝑦𝑡
𝜎2Know noise variance,
Received signal,
Sample covariance matrix
(estimated)
MMSE filter (estimated)
ℎ = 𝐹 𝐻 𝑾 𝑭𝑬 𝑪 𝐹𝑦
Estimated channel
Fast MMSE Estimator:
Generated channel, 𝑾 𝑭𝑬 = 𝐶𝑐(𝐶𝑐 + 𝛴)−1𝐶 𝛿 = 𝑐𝑜𝑣(ℎ 𝑡)
Channel covariance matrix MMSE filter coefficients
ℎ 𝑡Generated channel, 𝑪 𝒄 = 𝑐𝑖𝑟𝑐(𝐶 𝛿)
Circulant approximation of
channel covariance matrix
𝑾 𝟎 = 𝐹𝑾 𝑭𝑬
FFT of filter
𝑪 =
𝟏
𝝈 𝟐
𝒕=𝟏
𝑻
𝐹𝒚 𝒕
𝟐
𝑊𝑆𝐸 𝑐𝑜𝑚𝑝𝑢𝑡𝑎𝑡𝑖𝑜𝑛 → 𝛰(𝑀log𝑀)

13. MMSE channel estimation
𝑾 𝑭𝑬 𝑪 =
𝑒𝑥𝑝(𝑡𝑟 𝐹 𝐻 𝑾 𝟎 𝐹 𝐶 + 𝑇𝑙𝑜𝑔 𝐼 − 𝑾 𝑭𝑬 ) 𝐹 𝐻 𝑾 𝟎 𝐹𝑝(𝛿) 𝑑𝛿
𝑒𝑥𝑝(𝑡𝑟 𝐹 𝐻 𝑾 𝟎 𝐹 𝐶 + 𝑇𝑙𝑜𝑔 𝐼 − 𝑾 𝑭𝑬 ) 𝑝(𝛿) 𝑑𝛿
𝑝(𝛿)Distribution of channel parameters,
𝑦𝑡
𝜎2Know noise variance,
Received signal,
Sample covariance matrix
(estimated)
MMSE filter (estimated)
ℎ = 𝐹 𝐻 𝑾 𝑭𝑬 𝑪 𝐹𝑦
Estimated channel
Motivation for Low-Complexity Neural Network:
Generated channel, 𝑾 𝑭𝑬 = 𝐶𝑐(𝐶𝑐 + 𝛴)−1𝐶 𝛿 = 𝑐𝑜𝑣(ℎ 𝑡)
Channel covariance matrix MMSE filter coefficients
ℎ 𝑡Generated channel, 𝑪 𝒄 = 𝑐𝑖𝑟𝑐(𝐶 𝛿)
Circulant approximation of
channel covariance matrix
𝑾 𝟎 = 𝐹𝑾 𝑭𝑬
FFT of filter
𝑪 =
𝟏
𝝈 𝟐
𝒕=𝟏
𝑻
𝐹𝒚 𝒕
𝟐
𝑭 𝑯
𝒅𝒊𝒂𝒈 𝑭𝒂 𝑭𝒙 = 𝒂 ∗ 𝒙

14. MMSE channel estimation
Learning Fast MMSE Estimator using a Convolutional Neural Network:
𝑦𝑡
𝜎2Know noise variance,
Received signal,
ℎ = 𝐹 𝐻 𝑾 𝑪𝑵𝑵 𝐹𝑦
Estimated channel
ℎ 𝑡Generated channel,
Conv ReLU Conv
Bias Bias
𝑾 𝑪𝑵𝑵
𝑦𝑡Generated channel,
Parameter Value
Epochs 10 000
Training batch size 600
Testing batch size 100
Learning rate (64/M)*1e-4
Convolutional layer 2
Hierarchical Training
1. Train CNN for M antennas
2. Interpolate kernel from length M to 2M
3. Use interpolated values of kernel from previous step
for initializing CNN when training for M antennas.

15. MMSE channel estimation
Learning Fast MMSE Estimator using a Convolutional Neural Network:
ℎ 𝑡Generated channel,
Conv ReLU Conv
Bias Bias
𝑾 𝑪𝑵𝑵
𝑦𝑡Generated channel,

16. MSE per antenna at an SNR of 0 dB for estimation from a single snapshot (T=1). Channel model with
one propagation path with uniformly distributed angle.

17. MSE per antenna at an SNR of 0 dB for estimation from a single snapshot (T=1). Channel model with
three propagation paths.

18. MSE per antenna for M = 64 antennas and for estimation from a single snapshot (T = 1). Channel
model with three propagation paths.

19. Thank You