Digital Signal Processing 2 Les 2: Inleiding 2

Prof. dr. ir. Toon van Waterschoot Faculteit Industriële Ingenieurswetenschappen ESAT – Departement Elektrotechniek KU Leuven, Belgium

Digital Signal Processing 2: Vakinhoud •  •  •  •  •  •  •  •  •  •  •  • 

Les 1: Inleiding 1 (Discrete signalen en systemen) Les 2: Inleiding 2 (Wiskundige concepten) Les 3: Spectrale analyse Les 4: Elementair filterontwerp Les 5: Schattingsproblemen Les 6: Lineaire predictie Les 7: Optimale filtering Les 8: Adaptieve filtering Les 9: Detectieproblemen Les 10: Classificatieproblemen Les 11: Codering Les 12: Herhalingsles

Les 2: Inleiding 2 •  Complex number theory complex numbers, complex plane, complex sinusoids, circular motion, sinusoidal motion, …

•  Signal transforms z-transform, Fourier transform, discrete Fourier transform

•  Matrix algebra vectors, matrices, linear systems of equations

Les 2: Literatuur •  Complex number theory J. O. Smith III, Mathematics of the DFT －  Ch. 2, “Introduction to Complex Numbers” －  Ch. 4, Section 4.2, “Complex Sinusoids”

•  Signal transforms S. J. Orfanidis, Introduction to Signal Processing －  Ch. 5, “z-Transforms” －  Ch. 9, “DFT/FFT Algorithms” M. H. Hayes, Statistical Digital Signal Processing and Modeling －  [summary] Ch. 2, Sections 2.2.4, 2.2.5, 2.2.8

•  Matrix algebra M. H. Hayes, Statistical Digital Signal Processing and Modeling －  Ch. 2, Section 2.3, “Linear Algebra”

Les 2: Inleiding 2 •  Complex number theory complex numbers, complex plane, complex sinusoids, circular motion, sinusoidal motion, …

•  Signal transforms z-transform, Fourier transform, discrete Fourier transform

•  Matrix algebra vectors, matrices, linear systems of equations

Complex number theory •  Complex numbers －  －  －  － 

roots of a quadratic polynomial equation fundamental theorem of algebra complex numbers complex plane

•  Complex sinusoids －  －  －  － 

complex numbers ª complex sinusoids circular motion positive and negative frequencies sinusoidal motion

Complex numbers (1)

complex numbers? “imaginary” roots of a polynomial equation

Complex numbers (2) •  roots of a quadratic polynomial equation: － 

consider a quadratic polynomial, describing a parabola:

p(x) = ax2 + bx + c (a > 0) － 

the roots of the polynomial correspond to the points where the parabola crosses the horizontal x-axis 2

ax + bx + c = 0 , x1,2 =

b±

p

b2 2a

4ac

Complex numbers (3)

p(x)

•  roots of a quadratic polynomial equation: － 

－ 

－ 

2

if b 4ac > 0 the polynomial has 2 real roots, and the parabola has 2 distinct intercepts with the x-axis if b2 4ac = 0 the polynomial has 1 real root (with multiplicity 2), and the parabola has 1 intercept (tangent point) with the x-axis 2 if b 4ac < 0 the polynomial has no real roots, and the parabola has no intercepts with the x-axis

x p(x)

x p(x)

x

Complex numbers (4) •  roots of a quadratic polynomial equation: － 

－ 

alternatively, if b2 4ac < 0 we could say that the polynomial has 2 “imaginary roots”, and the parabola has 2 “imaginary” intercepts with the x-axis these imaginary roots are represented as complex numbers:

x1,2 =

b±j

with

j,

p

p

(b2 2a

4ac) p(x)

1 x

Complex numbers (5) fundamental theorem of algebra:

every n-th order polynomial has exactly n complex roots

p(x)

= =

an xn + an 1 xn n Y an (x xi )

1

+ . . . + a1 x + a0

i=1

ai 2 R, x, xi 2 C

Complex numbers (6) •  complex numbers: － 

complex number:

x = |{z} a + jb |{z} Re

－ 

complex conjugate:

－ 

modulus:

－ 

argument:

p

x ¯=a

with a, b 2 R

Im

jb

p

a2 + b2 = x¯ x ✓ ◆ b \x = arctan a

|x| =

•  complex field: － 

the complex numbers form a field, and all algebraic rules for real numbers also apply for complex numbers

Complex numbers (7) •  complex plane: x = a + jb － 

the modulus and argument naturally lead to a radial representation in the complex plane

p r , |x| = a2 + ✓ b2 ◆ b ✓ , \x = arctan a Im

jb

complex plane

r

a = r cos ✓ b = r sin ✓

✓ a

Re

Complex number theory •  Complex numbers －  －  －  － 

roots of a quadratic polynomial equation fundamental theorem of algebra complex numbers complex plane

•  Complex sinusoids －  －  －  － 

complex numbers ª complex sinusoids circular motion positive and negative frequencies sinusoidal motion

Complex sinusoids (1) •  complex variable ➙ complex sinusoid: － 

from the radial representation we obtain

x = a + jb = r(cos ✓ + j sin ✓) － 

replacing

✓ = !k

x[k] = r(cos !k + j sin !k) － 

using Euler’s identity we get

x[k] = rej!k = r(cos !k + j sin !k)

Complex sinusoids (2) •  circular motion: － 

a complex sinusoid can be seen as a vector which describes a circular trajectory in the z-plane

x[k] = rej!k = r(cos !k + j sin !k) Im

z-plane

r

rej!k !k

r r

r

Re

Complex sinusoids (3) •  positive and negative frequencies: － 

－ 

for positive frequencies ! > 0 the circular motion is in counterclockwise direction for negative frequencies ! < 0 the circular motion is in clockwise direction Im

!>0

r

re !k

r r

Im

z-plane

r

r

j!k

Re

z-plane

rej!k !k

r r

r

Re

Complex sinusoids (4) •  sinusoidal motion: － 

sinusoidal motion is the projection of circular motion onto any straight line in the z-plane, e.g., j!k •  r cos !k is the projection of re onto the Re-axis •  r sin !k is the projection of rej!k onto the Im-axis

Im

Re

k

Les 2: Inleiding 2 •  Complex number theory complex numbers, complex plane, complex sinusoids, circular motion, sinusoidal motion, …

•  Signal transforms z-transform, Fourier transform, discrete Fourier transform

•  Matrix algebra vectors, matrices, linear systems of equations

Signal transforms •  z-transform －  －  － 

definition & properties complex variables region of convergence

•  Fourier transform －  － 

frequency response Fourier transform

•  Discrete Fourier Transform (DFT)

－ 

definition inverse DFT matrix form properties

－ 

Fast Fourier Transform (FFT)

－ 

Digital filtering using the DFT/FFT

－  －  － 

z-transform (1) •  definition:

discrete-time sequence in integer variable k z-transform Z(·) discrete-time series in complex variable z

z-transform (2) •  definition: － 

z-transform of a discrete-time signal:

{x[k]} = {. . . , x[ k], . . . , x[ 1], x[0], x[1], . . . , x[k], . . .}

z-transform Z(·)

X(z) = . . . + x[ k]z k + . . . + x[ 1]z + x[0] +x[1]z Z(x[k]) = X(z) =

1

+ . . . + x[k]z

1 X

k= 1

x[k]z

k

k

+ ...

z-transform (3) •  definition: － 

z-transform of a discrete-time system impulse response:

{h[k]} = {. . . , h[ k], . . . , h[ 1], h[0], h[1], . . . , h[k], . . .}

z-transform Z(·)

H(z) = . . . + h[ k]z k + . . . + h[ 1]z + h[0] +h[1]z Z(h[k]) = H(z) =

1

+ . . . + h[k]z

1 X

k= 1

h[k]z

k

k

+ ...

z-transform (4) •  properties: － 

linearity property:

⇢ － 

Z(ax[k]) = aX(z) Z(x1 [k] + x2 [k]) = X1 (z) + X2 (z)

time-shift theorem:

Z(x[k － 

d]) = z

d

X(z)

convolution theorem:

y[k] = h[k] ⇤ x[k] ) Y (z) = H(z)X(z)

z-transform (5) •  region of convergence: － 

the z-transform of an infinitely long sequence is a series with an infinite number of terms

X(z) = . . . + x[ k]z k + . . . + x[ 1]z + x[0] +x[1]z －  － 

1

+ . . . + x[k]z

k

+ ...

for some values of z the series may not converge the z-transform is only defined within the region of convergence (ROC):

ROC = {z 2 C|X(z) =

1 X

k= 1

x[k]z

k

6= 1}

Signal transforms •  z-transform －  －  － 

definition & properties complex variables region of convergence

•  Fourier transform －  － 

frequency response Fourier transform

•  Discrete Fourier Transform (DFT)

－ 

definition inverse DFT matrix form properties

－ 

Fast Fourier Transform (FFT)

－ 

Digital filtering using the DFT/FFT

－  －  － 

Fourier transform (1) •  Frequency response: － 

for an LTI system a sinusoidal input signal

x[k] = ej!k = (cos !k + j sin !k) produces a sinusoidal output signal at the same frequency

y[k] = Aej(!k+ － 

)

= A cos(!k + ) + j sin(!k + )

the output can be calculated from the convolution:

y[k]

=

1 X

h[m]x[k

m] =

m= 1

=

ej!k

1 X

m= 1

1 X

m= 1

h[m]e

j!m

h[m]ej!(k

m)

Fourier transform (2) •  Frequency response: 1 X h[m]e j!m = m= 1

－  － 

1 X

h[m]z

m z=ej!

m= 1

=

H(z)

=

H(e )

z=ej! j!

the sinusoidal I/O relation is Aej(!k+ ) = H(ej! )ej!k the system’s frequency response H(ej! ) is a complex function of the radial frequency ω: •  • 

|H(ej! )| denotes the magnitude response \H(ej! ) denotes the phase response

Fourier transform (3) •  Frequency response: － 

－ 

the frequency response H(ej! ) is equal to the z-transform of the system’s impulse response, evaluated at z = ej! for 0  ! < 2⇡, ej! is a complex function describing the unit circle in the z-plane Im 1

e

z-plane j!

! -1

1 -1

Re

Fourier transform (4) •  Frequency response & Fourier transform –  the frequency response H(ej! ) of an LTI system is equal to the Fourier transform of the continuous-time impulse sequence constructed with h[k] :

F{hD (t)} = F

(

1 X

h[k] (t

kTs )

k= 1

)

f = . . . = H(e ), ! = 2⇡ fs j!

j!

j!

–  the frequency spectrum of a discrete-time signal X(e ), Y (e ) (=its z-transform evaluated at the unit circle) is similarly equal to the Fourier transform of the continuous-time impulse sequence constructed with x[k], y[k]:

F{xD (t)} = F

(

1 X

k= 1

x[k] (t

•  Input/output relation:

kTs )

)

f = . . . = X(e ), ! = 2⇡ fs j!

Y (ej! ) = H(ej! )X(ej! )

Signal transforms •  z-transform －  －  － 

definition & properties complex variables region of convergence

•  Fourier transform －  － 

frequency response Fourier transform

•  Discrete Fourier Transform (DFT)

－ 

definition inverse DFT matrix form properties

－ 

Fast Fourier Transform (FFT)

－ 

Digital filtering using the DFT/FFT

－  －  － 

Discrete Fourier Transform (DFT) (1) •  DFT definition: － 

the Fourier transform of a signal or system is a continuous function of the radial frequency ω:

F(x[k]) = X(ej! ) = － 

1 X

x[k]e

j!k

k= 1

the Fourier transform can be discretized by sampling it at N

2⇡ discrete frequencies !n = n, uniformly spaced between N 0 and 2π:

h

X e

j 2⇡ N n

i

=

N X1 k=0

x[k]e

j 2⇡ N nk

= DFT

Discrete Fourier Transform (DFT) (2) •  Inverse discrete Fourier transform (IDFT): － 

an N-point DFT can be calculated from an N-point time sequence:

h

X e － 

j 2⇡ N n

i

=

N X1

x[k]e

j 2⇡ N nk

k=0

vice versa, an N-point time sequence can be calculated from an N-point DFT:

1 x[k] = N

N X1 k=0

h

X e

j

2⇡ N n

i

e

j 2⇡ N nk

= IDFT

Discrete Fourier Transform (DFT) (3) •  matrix form － 

using the shorthand notations

(

X[n] WN

h

=

X ej

=

j 2⇡ N

e

2⇡ N n

i

the DFT and IDFT definitions can be rewritten as:

DFT:

X[n] =

N X1

x[k]WNnk

k=0

IDFT:

N 1 1 X x[k] = X[n]WN nk N n=0

Discrete Fourier Transform (DFT) (4) •  matrix form － 

2 6 6 6 4

the DFT coefficients X[0], . . . , X[N calculated as

X[0] X[1] .. . X[N

3

2

WN0 6W 0 6 N

7 7 6 7 = 6 .. 5 4 . 1] WN0

WN0 WN1 .. .

(N WN

1)

1] can then be

... ... .. .

WN0 (N 1) WN

...

(N WN

.. .

1)2

32 76 76 76 74 5

x[0] x[1] .. . x[N

X = WN x － 

3

an N-point DFT requires N2 complex multiplications

1]

7 7 7 5

Discrete Fourier Transform (DFT) (5) •  matrix form － 

2 6 6 6 4

x[0] x[1] .. . x[N

the IDFT coefficients x[0], . . . , x[N calculated as

3

2

WN0 6W 0 6 N

7 1 6 7 7= .. 5 N6 4 . 1] WN0

WN0 WN 1 .. .

WN

1 ⇤ x = WN X N

－ 

(N

1)

1] can then be

... ... .. .

WN0 (N 1) WN

...

(N

.. .

WN

1)2

32 76 76 76 74 5

3

X[0] X[1] .. . X[N

an N-point IDFT requires N2 complex multiplications

1]

7 7 7 5

Discrete Fourier Transform (DFT) (6) •  properties: －  － 

linearity & time-shift theorem (cf. z-transform) frequency-shift theorem (modulation theorem):

e － 

j!ck

x[k]

! X[n + c]

circular convolution theorem: if x[k] and h[k] are periodic with period N, then

N X1 m=0

x[m]h[k

m] =

N X1 m=0

x[k

m]h[m]

! X[n]H[n]

Discrete Fourier Transform (DFT) (7) •  Fast Fourier Transform (FFT)

Carl Friedrich Gauss (1777-1855

•  •  •  •  •  •  •  •  •  － 

split up N-point DFT in two N/2-point DFTs split up two N/2-point DFT’s in four N/4-point DFTs … split up N/2 2-point DFT’s in N 1-point DFTs calculate N 1-point DFTs rebuild N/2 2-point DFTs from N 1-point DFTs … rebuild two N/2-point DFTs from four N/4-point DFTs rebuild N-point DFT from two N/2-point DFTs

DFT complexity of N2 multiplications is reduced to FFT complexity of 1/2 N log2(N) multiplications

James W. Cooley

divide-and-conquer approach:

John W.Tukey

－ 

Discrete Fourier Transform (DFT) (8) •  Linear and circular convolution: － 

circular convolution theorem: due to the sampling of the frequency axis, the IDFT of the product of two N-point DFTs corresponds to the circular convolution of two length-N periodic signals N X1

x[m]h[k

m] =

m=0

－ 

N X1

x[k

m]h[m]

m=0

! X[n]H[n]

LTI system: the output sequence is the linear convolution of the impulse response with the input signal

y[k] =

N X1 m=0

x[k

m]h[m]

Discrete Fourier Transform (DFT) (9) •  Linear and circular convolution: － 

the linear convolution of a length-(M+1) impulse response h[k] with a length-L input signal x[k] is equivalent to their N-point circular convolution if both sequences are zeropadded to length N:

N h[k] x[k] y[k]

L+M

zero padding

Les 2: Inleiding 2 •  Complex number theory complex numbers, complex plane, complex sinusoids, circular motion, sinusoidal motion, …

•  Signal transforms z-transform, Fourier transform, discrete Fourier transform

•  Matrix algebra vectors, matrices, linear systems of equations

Matrix algebra: overview •  Vectors definition & geometrical interpretation －  elementary operations －  inner product & angle －  norm －  outer product －  linear (in)dependence •  Matrices •  Linear systems of equations － 

Vectors: definition & geom. interpretation •  Definition array of real- or complex-valued numbers or variables (in DSP: array of signal samples, DFT coefficients, …) －  vector transpose: column vector ⟷ row vector － 

3

2

x1 6 x2 7 ⇥ 6 7 T x = 6 . 7 , x = x1 4 .. 5 xN

•  Geometrical interpretation － 

array of coordinates of a point in N-dimensional space

x2 x2

...

xN

⇤

2-D example: x=

1 2

x1



2 1

Vectors: elementary operations •  Addition







x1 y1 x 1 + y1 x+y = + = =z x2 y2 x 2 + y2 z

y x

•  Scalar product － 

changes vector length but not direction





x1 ax1 ax = a = x2 ax2

ax x

Vectors: inner product & angle •  Inner product (= scalar !) T

⇥

hx, yi = x y = x1

x2

...

•  Angle －  － 

－ 

2

3

y1 N 7 X y ⇤6 6 27 xN 6 . 7 = x n yn 4 .. 5 n=1

yN

y

↵ let ||x|| denote length of vector x x angle between vectors is then related to inner product

hx, yi = kxkkyk cos(↵)

orthogonal vectors have zero inner product

Vectors: norm v uN uX p p kxk = kxk2 = hx, xi = xT x = t |xn |2

•  Norm = Euclidian norm = L2 norm •  Geometric interpretation

n=1

vector length: kxk －  distance between vectors: kx yk •  Widely used in DSP ! －  norm → signal RMS value －  squared norm → average signal power •  Other norms X N L1 norm: kxk1 = |xn |, L1 norm: kxk1 = max |xn | － 

n=1

n

Vectors: outer product •  Outer product (= matrix !)

xy

T

=

=

3

2

x1 6 x2 7 ⇥ 6 7 6 .. 7 y1 4 . 5 2

y2

...

xN

x 1 y1 6 x 2 y1 6 6 .. 4 .

x N y1

x 1 y2 x 2 y2 .. .

... ... .. .

x N y2

...

yN

⇤ 3

x 1 yN x 2 yN 7 7 .. 7 . 5

x N yN

Vectors: linear (in)dependence •  Linear (in)dependence －  － 

property of a set of vectors set of P vectors x1 , x2 , . . . , xP is linearly independent: P X

n=1 － 

－ 

an x n = 0 ) a1 = a2 = . . . = aP = 0

size P of set of linearly independent vectors ≤ length N trivial

2 3 2 3 2 3 0 0 1 607 607 617 6 7 7 6 7 example: 6 6 .. 7 , 6 .. 7 , . . . , 6 .. 7 4.5 4.5 4.5 0

0

1

•  Related concepts: space, basis, dimension

Matrix algebra: overview •  Vectors •  Matrices definition & elementary operations －  matrix product －  matrix-vector product －  matrix decomposition & rank －  structured matrices －  matrix form convolution •  Linear systems of equations － 

Matrices: definition & elementary ops. •  Definition: M x N matrix

2

a11 6 a21 6 A = {am,n } = 6 . 4 ..

aM 1

a12 a22 .. .

... ... .. .

a1N a2N .. .

aM 2

...

aM N

•  Elementary operations: －  －  － 

transpose: AT = {an,m }

(= N x M matrix)

addition: A + B = {am,n + bm,n } scalar product: cA = {cam,n }

3 7 7 7 5

Matrices: matrix product •  Matrix product － 

dimensions:

A |{z} B = |{z} C |{z}

M ⇥N N ⇥P － 

definition: C = {cm,p } =

•  Example 

a11 a21

a12 a22

a13 a23

2

b11 4b21 b31

b12 b22 b32

b13 b23 b33

M ⇥P

(

N X

am,n bn,p

n=1

3

 b14 c b24 5 = 11 c21 b34

)

c12 c22

c13 c23

c14 c24

Matrices: matrix-vector product •  Matrix-vector product: •  Two interpretations: － 

A |{z} b = |{z} c |{z}

M ⇥N N ⇥1

M ⇥1

matrix-vector product = stacking inner products

⇥

am1

am2

...

⇤

amN b = cm , 8m

product = linear combination of columns 3 2 3 3 2 3 2 2matrix-vector c1 a1N a12 a11 6 a2N 7 6 c2 7 6 a22 7 6 a21 7 6 7 6 7 6 7 7 6 b 1 6 . 7 + b2 6 . 7 + . . . + b N 6 . 7 = 6 . 7 4 .. 5 4 .. 5 4 .. 5 4 .. 5 － 

aM 1

aM 2

aM N

cM

Matrices: matrix decomposition & rank •  Matrix decomposition － 

N x N matrix can be decomposed as sum of R ≤ N outer products of linearly independent vectors

A=

R X

T v v n n n

n=1

－ 

“eigenvalue decomposition”: •  eigenvalues n •  eigenvectors vn

general M x N matrices: “singular value decomposition” •  Rank = R －  rank-deficient / singular matrix: R < min(M, N ) －  full-rank matrix: R = min(M, N ) － 

Matrices: structured matrices (N x N) •  Diagonal matrices & identity matrices

2

a11 6 0 6 A=6 . 4 .. 0

3

0 a22 .. .

... ... .. .

0 0 .. .

0

...

aN N

•  Symmetric matrices:

2

a11 6 a12 6 T A=A =6 . 4 ..

a1N

7 7 7, 5

a12 a22 .. .

... ... .. .

a2N

...

2

1 60 6 I = 6. 4 .. 0

3

a1N a2N 7 7 .. 7 . 5

aN N

0 1 .. .

... ... .. .

0

...

3 0 07 7 .. 7 .5 1

Matrices: structured matrices (M x N) •  Hankel: constant anti-diagonals (always symmetric !) 2

a1 6 a2 6 6 A = 6 a3 6 .. 4 .

aM

a2 a3 a4 .. .

a3 a4 a5 .. .

... ... ... .. .

aM +1

aM +2

...

3

aN aN +1 aN +2 .. . aN +M

1

7 7 7 7 7 5

•  Toeplitz & circulant: constant diagonals 2

aM 6aM 1 6 6 A = 6aM 2 6 .. 4 . a1

aM +1 aM aM 1 .. .

aM +2 aM +1 aM .. .

... ... ... .. .

aM +N aM +N aM +N .. .

a2

a3

...

aN

3 2

a1 7 6 2 7 6 a2 7 6 3 7 , 6 a3 7 6 .. 5 4 .

1

aM

aN a1 a2 .. . aM +N

aN 1 aN a1 .. . 1

aM +N

2

... ... ... .. .

a2 a3 a4 .. .

...

aM +1

3 7 7 7 7 7 5

Matrices: matrix form convolution •  Input signal vector or Toeplitz matrix 2

6 6 6 x=6 6 4

3

x0 x1 x2 .. . xL

1

7 7 7 7 7 5

2

x0 6x1 6 6 X = 6 x2 6 .. 4.

0 x0 x1 .. .

0 0 x0 .. .

... ... ... .. .

0 0 0 .. .

0

0

...

xL

0

•  Filter vector or Toeplitz matrix 3

2

h0 6 h1 7 6 7 6 7 h = 6 h2 7 6 .. 7 4 . 5 hM

2

h0 6 h1 6 6 H = 6 h2 6 .. 4.

0 h0 h1 .. .

0 0 h0 .. .

... ... ... .. .

0 0 0 .. .

0

0

...

hM

0

•  Output signal vector:

3

1

7 7 7 7 7 5

3 7 7 7 7 7 5

y = Hx = Xh

Matrix algebra: overview •  Vectors •  Matrices •  Linear systems of equations －  － 

square systems + matrix inverse rectangular systems + matrix pseudo-inverse

Linear systems of equations: square •  Square systems: no. equations = no. unknowns = N a11 x1 + a12 x2 + . . . + a1N xN

=

b1

a21 x1 + a22 x2 + . . . + a2N xN

= .. .

b2

aN 1 x1 + aN 2 x2 + . . . + aN N xN

=

bN

, Ax = b

•  Solution vector

x=A

1

b

•  Matrix inverse A

1

exists only if matrix rank R = N

Linear systems of equations: rectangular •  Rectangular systems: M equations, N unknowns a11 x1 + a12 x2 + . . . + a1N xN

=

b1

a21 x1 + a22 x2 + . . . + a2N xN

= .. .

b2

aM 1 x1 + aM 2 x2 + . . . + aM N xN

=

bM

, Ax = b

•  Underdetermined systems: M < N －  － 

infinitely many solutions ! minimum-norm solution: min kxk s.t. Ax = b

x=A |

T

AA {z A+

T

1

}

b

Linear systems of equations: rectangular •  Overdetermined systems: M > N －  － 

no exact solution ! best approximation (in least squares sense): T

x= A A | {z

A+

•  Pseudo-inverse －  － 

1

AT b }

underdetermined case A overdetermined case A

+

+

T

AA

T

1

=A

= A A

T

AT

1