diagbfweights

Diagonalize MIMO channel

Syntax

[wp,wc] = diagbfweights(chanmat)

[wp,wc,P] = diagbfweights(chanmat)

[wp,wc,P,G] = diagbfweights(chanmat)

[wp,wc,P,G,C] = diagbfweights(chanmat)

[___] = diagbfweights(chanmat,Pt)

[___] = diagbfweights(chanmat,Pt, Pn)

[___] = diagbfweights(chanmat,Pt, Pn,powdistoption)

Description

[wp,wc] = diagbfweights(chanmat) returns precoding weights, wp, and combining weights, wc, for the channel response matrix, chanmat. Together, these weights diagonalize the channel into subchannels so that the matrix wp*chanmat*wc is diagonal.

[wp,wc,P] = diagbfweights(chanmat) also returns the distributed power, P, for each element of the transmitting array.

[wp,wc,P,G] = diagbfweights(chanmat) also returns the subcarrier gains, G.

[wp,wc,P,G,C] = diagbfweights(chanmat) also returns the channel capacity sum, C.

[___] = diagbfweights(chanmat,Pt) also specifies total transmit power, Pt, and returned values any of the previous output argument combinations.

[___] = diagbfweights(chanmat,Pt, Pn) also specifies the noise power per transmitting antenna, Pn.

[___] = diagbfweights(chanmat,Pt, Pn,powdistoption) also specifies the noise distribution, powdistoption, across all transmitting antennas.

Examples

Compute and Diagonalize Channel Matrix

Compute the channel matrix for a 4-by-4 transmitting URA array and a 5-by-5 receiving URA array. Assume that three scatterers are randomly located within a specified angular range. The element spacing for both arrays is one-half wavelength. The receive array is 500 wavelengths away from the transmitting array along the x-axis. Constrain the angular span for the transmitting and receiving arrays. Diagonalize the channel matrix to compute the precoding and combining weights.

Specify the 4-by-4 transmitting array. Element spacing is in units of wavelength.

Nt = 4;
sp = 0.5;
ygridtx = (0:Nt-1)*sp - (Nt-1)/2*sp;
zgridtx = (0:Nt-1)*sp - (Nt-1)/2*sp;
[ytx,ztx] = meshgrid(ygridtx,zgridtx);
txpos = [zeros(1,Nt*Nt);ytx(:).';ztx(:).'];

Specify the 5-by-5 receiving array. Element spacing is in units of wavelength.

Nr = 5;
sp = 0.5;
ygridrx = (0:Nr-1)*sp - (Nr-1)/2*sp;
zgridrx = (0:Nr-1)*sp - (Nr-1)/2*sp;
[yrx,zrx] = meshgrid(ygridrx,zgridrx);
rxpos = [500*ones(1,Nr*Nr);yrx(:).';zrx(:).'];

Set the angular limits for transmitting and receiving.

• The azimuth angle limits for the transmitter are −45° to +45°.

• The azimuth angle limits for the receiver are −75° to +50°.

• The elevation angle limits for the transmitter are −12° to +12°.

• The elevation angle limits for the receiver are −30° to +30°.

angrange = [-45 45 -75 50; -12 12 -30 30];

Specify three scatterers and create the channel matrix.

numscat = 3;
chmat = scatteringchanmtx(txpos,rxpos,numscat,angrange);

Diagonalize the channel matrix.

[wp,wc] = diagbfweights(chmat);
z = wp*chmat*wc;

Show the first four diagonal elements.

z(1:4,1:4)

ans = 4×4 complex

  23.3713 - 0.0000i  -0.0000 - 0.0000i  -0.0000 + 0.0000i  -0.0000 + 0.0000i
   0.0000 - 0.0000i  10.7803 + 0.0000i  -0.0000 + 0.0000i   0.0000 + 0.0000i
  -0.0000 - 0.0000i  -0.0000 + 0.0000i   1.0566 + 0.0000i  -0.0000 - 0.0000i
  -0.0000 - 0.0000i   0.0000 + 0.0000i   0.0000 - 0.0000i   0.0000 + 0.0000i

Distributed Power of Diagonalized Channel Matrix

Compute the channel matrix for a 4-by-4 transmitting URA array and a 5-by-5 receiving URA array. Assume that three scatterers are randomly located within a specified angular range. The element spacings for both arrays is one-half wavelength. The receive array is 500 wavelengths away along the x-axis. Diagonalize the channel matrix to compute the precoding and combining weights and the distributed power.

Specify the 4-by-4 transmitting array. Element spacing is in units of wavelength.

Nt = 4;
sp = 0.5;
ygridtx = (0:Nt-1)*sp - (Nt-1)/2*sp;
zgridtx = (0:Nt-1)*sp - (Nt-1)/2*sp;
[ytx,ztx] = meshgrid(ygridtx,zgridtx);
txpos = [zeros(1,Nt*Nt);ytx(:).';ztx(:).'];

Specify the 5-by-5 receiving array. Element spacing is in units of wavelength.

Nr = 5;
sp = 0.5;
ygridrx = (0:Nr-1)*sp - (Nr-1)/2*sp;
zgridrx = (0:Nr-1)*sp - (Nr-1)/2*sp;
[yrx,zrx] = meshgrid(ygridrx,zgridrx);
rxpos = [500*ones(1,Nr*Nr);yrx(:).';zrx(:).'];

Set the angular limits for transmitting and receiving.

• The azimuth angle limits for the transmitter are −45° to +45°.

• The azimuth angle limits for the receiver are −75° to +50°.

• The elevation angle limits for the transmitter are −12° to +12°.

• The elevation angle limits for the receiver are −30° to +30°.

angrange = [-45 45 -75 50; -12 12 -30 30];

Specify three scatterers and create the channel matrix.

numscat = 3;
chmat = scatteringchanmtx(txpos,rxpos,numscat,angrange);

Diagonalize the channel matrix and return the distributed power.

[wp,wc,P] = diagbfweights(chmat);
disp(P.')

    0.0625
    0.0625
    0.0625
    0.0625
    0.0625
    0.0625
    0.0625
    0.0625
    0.0625
    0.0625
    0.0625
    0.0625
    0.0625
    0.0625
    0.0625
    0.0625

Subchannel Gains of Diagonalized Channel Matrix

Compute the channel matrix for an 11-element transmitting ULA array and a 7-element receiving ULA array. Assume that there are five randomly located scatterers. The element spacings for both arrays is one-half wavelength. The receive array is 500 wavelengths away from the transmit array along the x-axis. Diagonalize the channel matrix to compute the precoding and combining weights, the distributed power, and the subchannel gains.

Specify the 11-element transmitting ULA array. Element spacing is in units of wavelength.

Nt = 11;
sp = 0.5;
txpos = (0:Nt-1)*sp - (Nt-1)/2*sp;

Specify the 7-element receiving ULA array. Element spacing is in units of wavelength.

Nr = 7;
sp = 0.5;
rxpos = (0:Nr-1)*sp - (Nr-1)/2*sp;
numscat = 5;
chmat = scatteringchanmtx(txpos,rxpos,numscat);

Diagonalize the channel matrix and return the subchannel gains.

[wp,wc,P,G] = diagbfweights(chmat);
disp(G.')

  221.8345
   56.8443
   47.6711
    0.8143
    0.0000
    0.0000
    0.0000

Channel Capacity Sum of Diagonalized Channel Matrix

Compute the channel matrix for an 11-element transmitting ULA array and a 7-element receiving ULA array. Assume that there are five randomly located scatterers. The element spacings for both arrays is one-half wavelength. The receive array is 500 wavelengths away from the transmitting array along the x-axis. Create a channel matrix with two subcarriers. Diagonalize the channel matrix to compute the precoding and combining weights, the distributed power, the subchannel gains, and the channel capacity sum.

Specify the 11-element transmitting ULA array. Element spacing is in units of wavelength.

Nt = 11;
sp = 0.5;
txpos = (0:Nt-1)*sp - (Nt-1)/2*sp;

Specify the 7-element receiving ULA array. Element spacing is in units of wavelength.

Nr = 7;
sp = 0.5;
rxpos = (0:Nr-1)*sp - (Nr-1)/2*sp;
numscat = 5;

Create two subcarriers.

chmat1 = scatteringchanmtx(txpos,rxpos,numscat);
chmat2 = scatteringchanmtx(txpos,rxpos,numscat);
chmat(1,:,:) = chmat1;
chmat(2,:,:) = chmat2;

Diagonalize the channel matrix and return the subchannel gains.

[wp,wc,P,G,C] = diagbfweights(chmat);
disp(C.')

    9.5466    9.3605

Diagonalize Channel Matrix with Specified Power

Compute the channel matrix for an 11-element transmitting ULA array and a 7-element receiving ULA array. Specify the total transmitted power at 1000. Assume that there are five randomly located scatterers. The element spacings for both arrays is one-half wavelength. The receive array is 500 wavelengths away from the transmitting array along the x-axis. Create a channel matrix with two subcarriers. Diagonalize the channel matrix to compute the precoding and combining weights, the distributed power, the subchannel gains, and the channel capacity sum.

Specify the 11-element transmitting ULA array. Element spacing is in units of wavelength.

Nt = 11;
sp = 0.5;
txpos = (0:Nt-1)*sp - (Nt-1)/2*sp;

Specify the 7-element receiving ULA array. Element spacing is in units of wavelength.

Nr = 7;
sp = 0.5;
rxpos = (0:Nr-1)*sp - (Nr-1)/2*sp;
numscat = 5;

Create two subcarriers.

chmat1 = scatteringchanmtx(txpos,rxpos,numscat);
chmat2 = scatteringchanmtx(txpos,rxpos,numscat);
chmat(1,:,:) = chmat1;
chmat(2,:,:) = chmat2;

Diagonalize the channel matrix and return the distributed power for both subcarriers.

Pt = 1000.0;
[wp,wc,P,G,C] = diagbfweights(chmat,Pt);
disp(P.')

   90.9091   90.9091
   90.9091   90.9091
   90.9091   90.9091
   90.9091   90.9091
   90.9091   90.9091
   90.9091   90.9091
   90.9091   90.9091
   90.9091   90.9091
   90.9091   90.9091
   90.9091   90.9091
   90.9091   90.9091

Diagonalize Channel Matrix with Specified Noise Power

Compute the channel matrix for an 11-element transmitting ULA array and a 7-element receiving ULA array. Specify the total transmitted power at 1000 and the transmitting antenna noise power at 100. Assume that there are five randomly located scatterers. The element spacings for both arrays is one-half wavelength. The receive array is 500 wavelengths away from the transmit array along the x-axis. Create a channel matrix with two subcarriers. Diagonalize the channel matrix to compute the precoding and combining weights, the distributed power, subchannel gains, and channel capacity sum.

Specify the 11-element transmitting ULA array. Element spacing is in units of wavelength.

Nt = 11;
sp = 0.5;
txpos = (0:Nt-1)*sp - (Nt-1)/2*sp;

Specify the 7-element receiving ULA array. Element spacing is in units of wavelength.

Nr = 7;
sp = 0.5;
rxpos = (0:Nr-1)*sp - (Nr-1)/2*sp;
numscat = 5;

Create two subcarriers.

chmat1 = scatteringchanmtx(txpos,rxpos,numscat);
chmat2 = scatteringchanmtx(txpos,rxpos,numscat);
chmat(1,:,:) = chmat1;
chmat(2,:,:) = chmat2;

Diagonalize the channel matrix and return the gain for both subcarriers.

Pt = 1000.0;
Pn = 100.0;
[wp,wc,P,G,C] = diagbfweights(chmat,Pt,Pn);
disp(G.')

  221.8345  119.7549
   56.8443  115.9814
   47.6711   24.9780
    0.8143    5.1025
    0.0000    0.0059
    0.0000    0.0000
    0.0000    0.0000

Diagonalize Channel Matrix Using Waterfill Power Distribution

Compute the channel matrix for an 11-element transmitting ULA array and a 7-element receiving ULA array. Specify the total transmitted power at 1000 and the transmitting antenna noise power at 100. Specify the transmitted power distribution as 'Waterfill'. Assume that there are five randomly located scatterers. The element spacing for both arrays is one-half wavelength. The receive array is 500 wavelengths away from the transmitting array along the x-axis. Create a channel matrix with two subcarriers. Diagonalize the channel matrix to compute the precoding and combining weights, the distributed power, the subchannel gains, and the channel capacity sum.

Specify the 11-element transmitting ULA array. Element spacing is in units of wavelength.

Nt = 11;
sp = 0.5;
txpos = (0:Nt-1)*sp - (Nt-1)/2*sp;

Specify the 7-element receiving ULA array. Element spacing is in units of wavelength.

Nr = 7;
sp = 0.5;
rxpos = (0:Nr-1)*sp - (Nr-1)/2*sp;
numscat = 5;

Create two subcarriers.

chmat1 = scatteringchanmtx(txpos,rxpos,numscat);
chmat2 = scatteringchanmtx(txpos,rxpos,numscat);
chmat(1,:,:) = chmat1;
chmat(2,:,:) = chmat2;

Diagonalize the channel matrix and return the gain for both subcarriers.

Pt = 1000.0;
Pn = 100.0;
[wp,wc,P,G,C] = diagbfweights(chmat,Pt,Pn,'Waterfill');
disp(G.')

  221.8345  119.7549
   56.8443  115.9814
   47.6711   24.9780
    0.8143    5.1025
    0.0000    0.0059
    0.0000    0.0000
    0.0000    0.0000

Input Arguments

chanmat — Channel response matrix

N_t-by-N_r complex-valued matrix | L-by-N_t-by-N_r complex-valued MATLAB^® array

Channel response matrix, specified as an N_t-by-N_r complex-valued matrix or an L-by-N_t-by-N_r complex-valued MATLAB array.

• N_t is the number of elements in the transmitting array.

• N_r is the number of elements in the receiving array.

• L is the number of subcarriers.

When chanmat is a MATLAB array containing subcarriers, each subcarrier is decomposed independently into subchannels.

Data Types: double | single
Complex Number Support: Yes

Pt — Total transmit power

1 (default) | positive scalar | L-element vector of positive values

Total transmit power, specified as a positive scalar or an L-element vector of positive values. Pt has the same units as the total distributed power, P.

Data Types: double | single

Pn — Noise power

1 (default) | positive scalar

Noise power in each receiving antenna, specified as a positive scalar. Pn has the same units as the total transmit power, Pt.

Data Types: double | single

powdistoption — Power distribution option

'Uniform' (default) | 'Waterfill'

Power distribution option, specified as 'Uniform' or 'Waterfill'. When powdistoption is 'Uniform', the transmit power is evenly distributed across all N_t channels. If powdistoption is 'Waterfill', the transmit power is distributed across the N_t channels using a waterfill algorithm.

Data Types: char

Output Arguments

wp — Precoding weights

N_t-by-N_t complex-valued matrix | L-by-N_t-by-N_t complex-valued MATLAB array

Precoding weights, returned as an N_t-by-N_t complex-valued matrix or an L-by-N_t-by-N_t complex-valued MATLAB array. Units are dimensionless.

Data Types: double | single
Complex Number Support: Yes

wc — Combining weights

N_r-by-N_r complex-valued matrix | L-by-N_r-by-N_r complex-valued MATLAB array

Combining weights, returned as an N_r-by-N_r complex-valued matrix or an L-by-N_r-by-N_r complex-valued MATLAB array. Units are dimensionless.

Data Types: double | single
Complex Number Support: Yes

P — Distributed power

1-by-N_t real-valued row vector | L-by-N_t real-valued matrix

Distributed power, returned as a vector or matrix.

• When chanmat is an N_t-by-N_r real-valued matrix, P is a 1-by-N_t real-valued row vector.

• When chanmat is an L-by-N_t-by-N_r real-valued MATLAB array, P is an L-by-N_t real-valued matrix.

Power units are linear.

Data Types: double | single

G — Subchannel gains

1-by-N_g complex-valued row vector | L-by-N_g complex-valued matrix

Subchannel gains, returned as a vector or matrix.

• When chanmat is an N_t-by-N_r complex-valued matrix, G is a 1-by-N_g complex-valued row vector.

• When chanmat is an L-by-N_t-by-N_r complex-valued MATLAB array, G is an L-by-N_g complex-valued matrix.

N_g is the smaller of N_t and N_r.

Gain units are linear.

Data Types: double | single
Complex Number Support: Yes

C — Channel capacity sum for each subcarrier

scalar | L-by-1 vector

Channel capacity sum for each subcarrier, returned as a scalar or vector.

• When chanmat is an N_t-by-N_r complex-valued matrix, C is a scalar.

• When chanmat is an L-by-N_t-by-N_r complex-valued MATLAB array, C is an L-by-1 vector.

Capacity units are in bps/Hz.

Data Types: double | single
Complex Number Support: Yes

References

[1] Heath, R. Jr. et al. “An Overview of Signal Processing Techniques for Millimeter Wave MIMO Systems”, arXiv.org:1512.03007 [cs.IT], 2015.

[2] Tse, D. and P. Viswanath, Fundamentals of Wireless Communications, Cambridge: Cambridge University Press, 2005.

[3] Paulraj, A. Introduction to Space-Time Wireless Communications, Cambridge: Cambridge University Press, 2003.

Extended Capabilities

C/C++ Code Generation

Generate C and C++ code using MATLAB® Coder™.

Usage notes and limitations:

• Does not support variable-size inputs.

Version History

Introduced in R2017a