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Signal Processing
Collection of Formulas
M. Dahl S. Johansson M. Winberg
October 26, 2005
Version 1.01
Department of Telecommunications and Signal Processing
Blekinge Institute of Technology
SE-372 33 Ronneby
Sweden
Contents
1 Basic Relationships and Concepts 31.1 Standard Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1.1 Continuous Functions . . . . . . . . . . . . . . . . . . . . . . 31.1.2 Discrete Time Functions . . . . . . . . . . . . . . . . . . . . 3
1.2 SISO systems (Single input, single output)Discrete Time Systems . . . . . . . . . . . . . . . . . . . . . . . . . 41.2.1 FIR Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2.2 IIR Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2.3 IIR Canonical form - Direct Form I . . . . . . . . . . . . . . 51.2.4 IIR Canonical form - Direkt Form II . . . . . . . . . . . . . 6
1.3 Some Methods of Calculation . . . . . . . . . . . . . . . . . . . . . 61.3.1 Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.3.2 State-Space Model of Difference Equation . . . . . . . . . . 61.3.3 State-Space Equation . . . . . . . . . . . . . . . . . . . . . . 71.3.4 System Function . . . . . . . . . . . . . . . . . . . . . . . . 7
1.4 Analogue Sinusoidal Signalthrough Linear, Causal Filter . . . . . . . . . . . . . . . . . . . . . 81.4.1 Complex, Non-causal Input Signal . . . . . . . . . . . . . . . 81.4.2 Complex, Causal Input Signal . . . . . . . . . . . . . . . . . 81.4.3 Real, Non-causal Input Signal . . . . . . . . . . . . . . . . . 81.4.4 Real, Causal Input Signal . . . . . . . . . . . . . . . . . . . 8
1.5 Discrete Time Sinusoidal Signalthrough Linear, Causal Filter . . . . . . . . . . . . . . . . . . . . . 101.5.1 Complex, Non-Causal Input Signal . . . . . . . . . . . . . . 101.5.2 Complex, Causal Input Signal . . . . . . . . . . . . . . . . . 101.5.3 Real, Non-Causal Input Signal . . . . . . . . . . . . . . . . . 101.5.4 Real, Causal Input Signal . . . . . . . . . . . . . . . . . . . 10
1.6 Fourier Series Expansion . . . . . . . . . . . . . . . . . . . . . . . . 121.6.1 Continuous Time . . . . . . . . . . . . . . . . . . . . . . . . 121.6.2 Discrete Time . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.7 Fourier Transformer . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.8 Discrete Fourier Transform (DFT) . . . . . . . . . . . . . . . . . . . 14
1.8.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.8.2 Circular Convolution . . . . . . . . . . . . . . . . . . . . . . 141.8.3 Non-Circular Convolution using DFT . . . . . . . . . . . . . 141.8.4 Relation to the Fourier Transform X(f): . . . . . . . . . . . 141.8.5 Relation to Fourier Series . . . . . . . . . . . . . . . . . . . 151.8.6 Parseval’s Theorem . . . . . . . . . . . . . . . . . . . . . . . 15
1.9 Some Window Functions and their Fourier Transforms . . . . . . . 16
2 Sampling Analogue Signals 182.1 Sampling and Reconstruction . . . . . . . . . . . . . . . . . . . . . 182.2 Measures of Distortion . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.2.1 Folding distortion when sampling . . . . . . . . . . . . . . . 202.2.2 Periodic Distortion when Reconstructing . . . . . . . . . . . 20
1
2.3 Quantizing Distortion . . . . . . . . . . . . . . . . . . . . . . . . . . 212.4 Decimation and Interpolation . . . . . . . . . . . . . . . . . . . . . 21
3 Analogue Filters 223.1 Filter Approximations if ideal LP filters . . . . . . . . . . . . . . . . 22
3.1.1 Butterworth Filters . . . . . . . . . . . . . . . . . . . . . . . 223.1.2 Chebyshev Filters . . . . . . . . . . . . . . . . . . . . . . . . 24
3.2 Frequency Transformations of Analogue Filter . . . . . . . . . . . . 29
4 Discrete Time Filters 304.1 FIR Filters and IIR Filters . . . . . . . . . . . . . . . . . . . . . . . 304.2 Construction of FIR Filter . . . . . . . . . . . . . . . . . . . . . . . 30
4.2.1 FIR Filter using the window method . . . . . . . . . . . . . 304.2.2 Equiripple FIR Filter . . . . . . . . . . . . . . . . . . . . . . 344.2.3 FIR Filters using Least-Squares . . . . . . . . . . . . . . . . 34
4.3 Constructing IIR Filters starting fromAnalogue Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354.3.1 The Impulse-Invariant Method . . . . . . . . . . . . . . . . . 354.3.2 Bilinear Transformation . . . . . . . . . . . . . . . . . . . . 354.3.3 Quantizing of Coefficients . . . . . . . . . . . . . . . . . . . 36
4.4 Lattice Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
5 Correlation 38
6 Spectrum Estimation 39
A Basic Relationships 43A.1 Trigonometrical formulas . . . . . . . . . . . . . . . . . . . . . . . . 43A.2 Some Common Relationships . . . . . . . . . . . . . . . . . . . . . 44A.3 Matrix Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
B Transforms 46B.1 The Laplace Transform . . . . . . . . . . . . . . . . . . . . . . . . . 46
B.1.1 The Laplace transform of causal signals . . . . . . . . . . . . 46B.1.2 One-sided Laplace transform of non-causal signals . . . . . . 47
B.2 The Fourier Transform of a Continuous Time Signal . . . . . . . . . 48B.3 The Z-transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
B.3.1 The Z-transform of causal signals . . . . . . . . . . . . . . . 50B.3.2 Single-Sided Z-transform of non-causal signals . . . . . . . . 51
B.4 Fourier Transform for Discrete Time Signal . . . . . . . . . . . . . . 52B.5 Some DFT Properties . . . . . . . . . . . . . . . . . . . . . . . . . 54
2
1 Basic Relationships and Concepts
1.1 Standard Signals
1.1.1 Continuous Functions
T is the period time for a perodic function f(t), where t [s] is the time variableAngular frequency Ω = 2π
T= 2πF [rad/s], F [Hz] is the frequency.
a) Impulse Function δ(t) =
+∞ t = 0
0 t = 0
∫∞−∞ δ(t)dt = 1
x(t)δ(t − a) = x(t − a)δ(t)
δ( 1a) = aδ(t)
∫∞−∞ x(t)δ(t − a)dt = x(a)
b) Step Function u(t) =
1 t ≥ 00 t < 0
c) Ramp Function r(t) =
t t ≥ 00 t < 0
d) Rectangular Pulse p(t) =
1 |t| < 12
0 |t| > 12
e) Sinc Function sinc t = sin πtπt
f) Periodic Function f(t) = f(t + T ) where T = 0
g) Periodic Sinc Function diric(t, T ) =sin
(Tt2
)T sin
(t2
)
h) Complex Periodic Signal est = eσtejΩt
i) Complex undamped per. signal ejΩt = cos Ωt + j sin Ωt
1.1.2 Discrete Time Functions
N is the period time (integer) for a periodic function f(n), where n is a discretetime index (integer). Normalized angular frequency ω = 2π
N= 2πf [rad], f is
normalized frequency.
3
a) Unit Pulse δ(n) =
1 n = 00 n = 0
b) Unit Step u(n) =
1 n ≥ 00 n < 0
c) Ramp r(n) =
n n ≥ 00 n < 0
d) Rectangular Pulse p(n) =
1 |n| < n1
0 |n| > −n1
e) Sinc Function sindN(n) =sin
(Nn2
)N sin
(n2
)
f) Periodic Signal f(n) = f(n + N) where N = 0
g) Complex periodic signal esn = eσnejωn
h) Complex undamped per. signal ejωn = cos ωn + j sin ωn
1.2 SISO systems (Single input, single output)Discrete Time Systems
1.2.1 FIR Filters
y(n) =M∑
k=0
bkx(n − k)
4
1.2.2 IIR Filters
y(n) = −N∑
k=1
aky(n − k)
1.2.3 IIR Canonical form - Direct Form I
y(n) = −N∑
k=1
aky(n − k) +M∑
k=0
bkx(n − k)
5
1.2.4 IIR Canonical form - Direkt Form II
+
z-1
v (n)N-1
z-1
+
+
+
+
1-a
-a 2
-a N-1
-a N
b1
b0
b2
bN-1
bN
z-1
vN (n)
v1(n)
+
+
x(n)
+
y(n)
y(n) = −N∑
k=1
aky(n − k) +M∑
k=0
bkx(n − k)
1.3 Some Methods of Calculation
1.3.1 Convolution
y(n) = h ∗ x =∞∑
k=−∞h(k)x(n − k) =
∞∑k=−∞
h(n − k)x(k)
1.3.2 State-Space Model of Difference Equation
If
y(n) = −N∑
k=1
aky(n − k) +M∑
k=0
bkx(n − k)
the State-Space Model is given byv(n + 1) = Fv(n) + q · x(n)
y(n) = gTv(n) + d · x(n)
where
F =
0 1 . . . 0 0...
......
...0 0 0 1
−ak −ak−1 . . . −a2 −a1
; q =
00...1
gT = (bk, . . . , b2, b1) − b0(ak, . . . , a2, a1) ; d = b0
6
1.3.3 State-Space Equation
a) Direct Solution
y(n) = gT · Fnv(0) +n−1∑k=0
gT · Fn−1−kqx(k)u(n − 1) + dx(n)
b) Impulse Function
h(n) = gT · Fn−1qu(n − 1) + dδ(n)
c) System FunctionH(z) = gT[zI − F]−1q + d
1.3.4 System Function
H(z) =Y(z)
X (z)=
b0 + b1z−1 + · · · + bMz−M
1 + a1z−1 + · · · + aNz−N
7
1.4 Analogue Sinusoidal Signalthrough Linear, Causal Filter
1.4.1 Complex, Non-causal Input Signal
x(t) = ejΩ0t = (cos(Ω0t) + j sin(Ω0t)) −∞ < t < ∞
y(t) =∫ ∞
τ=0h(τ)x(t − τ)dτ =
∫ ∞
τ=0h(τ)ejΩ0(t−τ)dτ = H(s)|s=jΩ0 ejΩ0t︸ ︷︷ ︸
stationary
1.4.2 Complex, Causal Input Signal
x(t) = ejΩ0tu(t) = (cos(Ω0t) + j sin(Ω0t)) u(t); X(s) =1
s − jΩ0
Y (s) = H(s)X(s) =T (s)
N(s)
1
s − jΩ0
=T1(s)
N(s)︸ ︷︷ ︸transient
+ H(s)|s=jΩ0
1
s − jΩ0︸ ︷︷ ︸stationary
y(t) = transient + H(s)|s=jΩ0 ejΩ0t︸ ︷︷ ︸stationary
1.4.3 Real, Non-causal Input Signal
x(t) = ReejΩ0t = cos(Ω0t) −∞ < t < ∞
y(t) =∫ ∞
τ=0h(τ)x(t − τ)dτ =
∫ ∞
τ=0h(τ)
1
2(ejΩ0(t−τ) + e−jΩ0(t−τ))dτ =
= |H(s)|s=jΩ0 cos(Ω0t + argH(s)|s=jΩ0)︸ ︷︷ ︸stationary
1.4.4 Real, Causal Input Signal
x(t) = ReejΩ0t u(t) = cos(Ω0t) u(t); X(s) =s
s2 + Ω20
Y (s) = H(s)X(s) =T (s)
N(s)
s
s2 + Ω20
=T1(s)
N(s)︸ ︷︷ ︸transient
+C1s + C0
s2 + Ω20︸ ︷︷ ︸
stationary
H(s)|s=jΩ0 = A ejθ; C1 = A cos(θ); C0 = −AΩ0 sin θ
8
y(t) = transient + C1 cos(Ω0t) +C0
Ω0
sin(Ω0t)︸ ︷︷ ︸stationary
=
= transient + |H(s)|s=jΩ0 cos(Ω0t + argH(s)|s=jΩ0)︸ ︷︷ ︸stationary
9
1.5 Discrete Time Sinusoidal Signalthrough Linear, Causal Filter
1.5.1 Complex, Non-Causal Input Signal
x(n) = ejω0n = (cos(ω0n) + j sin(ω0n)) −∞ < n < ∞
y(n) =∞∑
k=0
h(k)x(n − k) =∞∑
k=0
h(k)ejω0(n−k) = H(z)|z=ejω0 ejω0n︸ ︷︷ ︸stationary
1.5.2 Complex, Causal Input Signal
x(n) = ejω0nu(n) = (cos(ω0n) + j sin(ω0n)) u(n); X(z) =1
1 − ejω0z−1
Y (z) = H(z)X(z) =T (z)
N(z)
1
1 − ejω0z−1=
T1(z)
N(z)︸ ︷︷ ︸transient
+ H(z)|z=ejω0
1
1 − ejω0z−1︸ ︷︷ ︸stationary
y(n) = transient + H(z)|z=ejω0 ejω0n︸ ︷︷ ︸stationary
1.5.3 Real, Non-Causal Input Signal
x(n) = Reejω0n = cos(ω0n) −∞ < n < ∞
y(n) =∞∑
k=0
h(k)x(n − k) =∞∑
k=0
h(k)1
2(ejω0(n−k) + e−jω0(n−k)) =
= |H(z)|z=ejω0 cos(ω0n + argH(z)|z=ejω0)︸ ︷︷ ︸stationary
1.5.4 Real, Causal Input Signal
x(n) = Reejω0n u(n) = cos(ω0n) u(n); X(z) =1 − cos ω0z
−1
1 − 2 cos ω0z−1 + z−2
Y (z) = H(z)X(z) =T (z)
N(z)
1 − cos ω0z−1
1 − 2 cos ω0z−1 + z−2=
T1(z)
N(z)︸ ︷︷ ︸transient
+C0 + C1z
−1
1 − 2 cos ω0z−1 + z−2︸ ︷︷ ︸stationary
10
H(z)|z=ejω0 = A ejθ; C0 = A cos(θ); C1 = −A(sin ω0 sin θ + cos ω0 cos θ)
y(n) = transient + C0 cos(ω0n) +C1 + C0 cos(ω0)
sin(ω0)sin(ω0n)︸ ︷︷ ︸
stationary
=
= transient + |H(z)|z=ejω0 cos(ω0n + argH(z)|z=ejω0)︸ ︷︷ ︸stationary
11
1.6 Fourier Series Expansion
For table see Appendix
1.6.1 Continuous Time
A periodic function with period T0 , i.e. f(t) = f(t − T0), can be expressed in theform of a series expansion according to
f(t) =∞∑
k=−∞ck ej2πkF0t
where
ck =1
T0
∫T0
f(t)e−j2πkF0tdt ; F0 =1
T0
If f(t) is real this can also be expressed as
f(t) = c0 + 2∞∑
k=1
|ck| cos(2πkF0t + θk) =
= a0 +∞∑
k=1
ak cos 2πkF0t − bk sin 2πkF0t
where
a0 = c0 =1
T0
∫T0
f(t)dt
ak = 2|ck| cos θk =2
T0
∫T0
f(t) cos(2πkF0t)dt
bk = 2|ck| sin θk =−2
T0
∫T0
f(t) sin(2πkF0t)dt
The power is given by (Parseval’s Relation)
P =1
T0
∫T0
|f(t)|2dt =∞∑
k=−∞|ck|2
In addition, for real signals
P = c20 + 2
∞∑k=1
|ck|2 = a20 +
1
2
∞∑k=1
(a2k + b2
k)
1.6.2 Discrete Time
A periodic function with the period N , i.e. f(n) = f(n − N), can be expressed asa series expansion according to
f(n) =N−1∑k=0
ck ej2πk n/N
12
where
ck =1
N
N−1∑n=0
f(n) e−j2πk n/N , k = 0, . . . , N − 1
The series expansion is often written using DTFS (Discrete-Time Fourier Series).If f(n) is real this can also be expressed as
f(n) = c0 + 2L∑
k=1
|ck| cos
(2π
kn
N+ θk
)=
= a0 +L∑
k=1
(ak cos
(2π
kn
N
)− bk sin
(2π
kn
N
))
where
a0 = c0
ak = 2|ck| cos(θk)
bk = 2|ck| sin(θk)
L =
N2
if N even
N−12
if N odd
The power is given by
P =1
N
N−1∑n=0
|f(n)|2 =N−1∑k=0
|ck|2
and the energy over one period is given by
EN =N−1∑n=0
|f(n)|2 = NN−1∑k=0
|ck|2
1.7 Fourier Transformer
Continuous time signal:
Xa(F ) =∫∞−∞ xa(t)e
−j2πFtdt
xa(t) =∫∞−∞ Xa(F )ej2πFtdF
Discrete time signal:
X(f) =∑∞
n=−∞ x(n)e−j2πfn
x(n) =∫ 1/2−1/2 X(f)ej2πfndf
13
1.8 Discrete Fourier Transform (DFT)
1.8.1 Definition
Xk = DFT (xn) =N−1∑n=0
xne−j2πnk/N k = 0, 1, . . . , N − 1 Transform
xn = IDFT (Xk) =1
N
N−1∑k=0
Xkej2πnk/N n = 0, 1, . . . , N − 1 Inversion
Note:N−1∑n=0
ej2π k − k0
N · n= N · δ(k − k0, (modulo N))
1.8.2 Circular Convolution
xn N yn =N−1∑=0
xyn−DFT←→ XkYk Circular convolution
where∑
denotes circular convolution. This means that the sequences xn and yn
should be repeated periodically before the summation. I.e. outside the intervaln = 0, 1, . . . , N − 1, xn−N = xn and yn−N = yn ( = integer) In other words, theindex is calculated modulo N . Circular convolution is also denoted x(n) ∗ y(n).
1.8.3 Non-Circular Convolution using DFT
If x(n) = 0 for n = [0, L − 1] and y(n) = 0 for n = [0,M − 1] then x ∗ y = 0 forn = [0, N − 1] where N ≥ L + M − 1.The convolution can be calculated as
x ∗ y =
x ©∗ y = IDFT(XkYk) n = 0, 1, . . . , N − 1
0 Else
whereXk = DFT(x(n))Yk = DFT(y(n))
1.8.4 Relation to the Fourier Transform X(f):
X(k/N) = Xk = DFT (x(n)) if x(n) = 0 for n = [0, N − 1]
X(k/N) = Xk = DFT (xp(n)) generally x(n) where xp(n) =∞∑
=−∞x(n − N)
14
1.8.5 Relation to Fourier Series
X
(k
N
)= Xk = DFT (x(n)) = N · ck
ifx(n) = xp(n), 0 ≤ n ≤ N − 1
where
xp(n) =N−1∑k=0
ckej2π nk
N −∞ < n < ∞
and
ck =1
N
N−1∑n=0
xp(n)e−j2π nkN k = 0, 1, . . . , N − 1
1.8.6 Parseval’s Theorem
N−1∑n=0
x(n)y∗(n) =1
N
N−1∑k=0
XkY∗(k)
15
1.9 Some Window Functions and their Fourier Transforms
i) The window functions are centered around the origin (odd filter length M),i.e. the functions are not equal to zero only for −(M −1)/2 ≤ n ≤ (M −1)/2
Rectangular window:
wrect(n) = 1
Wrect(f) = M · sin(πfM)
M sin(πf)
Bartlett window (Triangular window):
w(n) = 1 − |n|(M − 1)/2
W (f) =M
2
(sin πfM
2M2
sin(πf)
)2
≈ 2
MW 2
rect
(f
2
)if f is small
Hanning Window:
w(n) = 0.5 + 0.5 cos(
2πn
M − 1
)
W (f) = 0.5 Wrect(f) +
+0.25 Wrect
(f − 1
M − 1
)+
+0.25 Wrect
(f +
1
M − 1
)
Hamming Window:
w(n) = 0.54 + 0.46 cos(
2πn
M − 1
)
W (f) = 0.54 Wrect(f) +
+0.23 Wrect
(f − 1
M − 1
)+
+0.23 Wrect
(f +
1
M − 1
)
Blackman window:
w(n) = 0.42 + 0.5 cos2πn
M − 1+ 0.08 cos
4πn
M − 1
W (f) = 0.42 Wrect(f) +
+0.25 Wrect
(f − 1
M − 1
)+
16
+0.25 Wrect
(f +
1
M − 1
)+
+0.04 Wrect
(f − 2
M − 1
)+
+0.04 Wrect
(f +
2
M − 1
)
Kaiser window:
w(n) =I0
(β√
1 − [2n/(M − 1)]2)
I0(β)
I0(x) = 1 +L∑
k=1
[(x/2)k
k!
]2
17
2 Sampling Analogue Signals
2.1 Sampling and Reconstruction
Sampling Theorem
If xa(t) is bandlimited, i.e. Xa(F ) = 0 for |F | ≥ 1/2T then
xa(t) =∞∑
n=−∞x(n)
sin πT
(t − nT )πT
(t − nT )
Sampling Frequency Fs = 1/T .
Sampling
x(n) = xa(nT ) ; T =1
Fs
X(f) = X(
F
Fs
)= Fs
∞∑k=−∞
Xa(F − kFs)
Γ(f) = Γ(
F
Fs
)= Fs
∞∑k=−∞
Γa(F − kFs)
Reconstruction (ideal)
xa(t) =∞∑
n=−∞x(n)
sin πT
(t − nT )πT
(t − nT )
Xa(F ) =1
Fs
X(
F
Fs
)|F | ≤ Fs
2
Γa(F ) =1
Fs
Γ(
F
Fs
)|F | ≤ Fs
2
Reconstruction using Sample-and-Hold
Xa(F ) =1
Fs
X(
F
Fs
)· sin(πFT )
πFTe−j2πF T
2 · HLP (F )
Γa(F ) =1
Fs
Γ(
F
Fs
) ∣∣∣∣∣sin(πFT )
πFT
∣∣∣∣∣2
· |HLP (F )|2
18
Block Scheme describing D/A conversion
Ideal reconstruction
-Fs/2 Fs/2
S d( t-n T)n ya (t)
|Y (F)|a
Lowpass
-Fs/2 Fs/2
S d( t-n T)n
ya (t)
ya (t)
|Y (F)|a
FFs|X( ) T |
x(n)
Tx(n)
x(1) T
| X(f)|
ya (t)
x(n)
Tx(n)
x(1) T
| X(f)|
n t t
f F F-.5 .5 -Fs/2 Fs/2 -Fs/2 Fs/2
ya(t) =∞∑
n=−∞x(n)
sin πT
(t − nT )πT
(t − nT )
Ya(F ) =1
Fs
X(
F
Fs
)|F | ≤ Fs
2
Reconstruction using Sample-and-Hold
Lowpass
-Fs/2 Fs/2
1
T
hSH (t) ya (t)
ya (t)
|Y (F)|aSH
|X( ) H (F)|Fs
F
S d( t-n T)n
x(n)
x(n)
| X(f)|
n t t
f F F-.5 .5 -Fs/2 Fs/2 -Fs/2 Fs/2
Ya(F ) =1
Fs
X(
F
Fs
)· sin(πFT )
πFTe−j2πF T
2 · HLP (F )
19
2.2 Measures of Distortion
2.2.1 Folding distortion when sampling
Spectrum after anti-aliasing filter:
Γin(F )
Folding distortion:
DA = 2 ·∫ ∞
Fs−Fp
Γin(F )dF
Effective signal power:
Ds = 2∫ Fp
0Γin(F )dF
where 0 ≤ Fp ≤ Fs/2
Signal distortion relationship:
A: SDRA = DS
DA=
∫ Fp
0Γin(F )dF∫∞
Fs−FpΓin(F )dF
B: SDR0A = min|F |≤Fp
Γin(F )Γin(Fs−F )
If the spectrum is monotonously decreasing
SDR0A =
Γin(Fp)
Γin(Fs − Fp)
2.2.2 Periodic Distortion when Reconstructing
Periodic Distortion:DP = 2 ·
∫ ∞
Fs/2Γut(F )dF
Effective signal power:
DS = 2 ·∫ Fs/2
0Γut(F )dF
Signal Distortion Relationship:
A: SDRP = DS
DP=
∫ Fs/2
0Γut(F )dF∫∞
Fs/2Γut(F )dF
B: SDR0P = min|F |<Fs/2
Γut(F )Γut(Fs−F )
A good estimate is often given by
SDR0P =
Γut(Fp)
Γut(Fs − Fp)
where Fp is the highest frequency component in the sampled signal.
20
2.3 Quantizing Distortion
DQ ∆2
12linear quantizing, ∆ small
SQNR =Signal Power
Dq
Quantizing distortion for sinusoidal signal, maximum dynamical range used, r bits
SQNR = 1.76 + 6 · r[dB]
Quantizing distortion, dynamical range usage expressed in top- and RMS value, rbits
SQNR = 6 · r + 1.76 − 1010 log
(Apeak
ARMS · √2
)2
− 1010 log
(V
Apeak
)2
where [−V, V ] is the dynamical range of the quantizer.
2.4 Decimation and Interpolation
Downsampling a factor M
↓ M y(n) = . . . u(0), u(M), u(2M) . . .
Y (f) = 1M
∑M−1i=0 U
(f−iM
)Upsampling a factor L
↑ L w(n) = . . . x(0), 0, 0, . . .︸ ︷︷ ︸L-1 st
, x(1), 0, 0, . . .︸ ︷︷ ︸L-1 st
, x(2) . . .
W (f) = X(fL)
21
3 Analogue Filters
3.1 Filter Approximations if ideal LP filters
General form of the amplitude function of the approximation
|H(Ω)| =K√
1 + gN
((ΩΩp
)2) Ω = 2πF
where
gN
(
Ω
Ωp
)2
1∣∣∣ ΩΩp
∣∣∣ < 1
1∣∣∣ ΩΩp
∣∣∣ > 1
and Ωp is the cut-off frequency of the filter.Sometimes it is suitable to normalize the angular frequency to Ωp.In this section this corresponds to setting Ωp = 1.
3.1.1 Butterworth Filters
|H(Ω)| =K√
1 +(
ΩΩp
)2N
The amplitude function always passes these points.
K = The maximum value of the amplitude function.K = The value of the amplitude functionens for Ω = 0.
Arbitrary attenuation in the pass band (X dB)
|H(Ω)| =K√
1 + ε2
(ΩΩ
′p
)2N
When Ω′p = Ωp (3dB) then ε2 ≈ 1.
22
The amplitude function always passes these points.
Any arbitraryattenuation
Filter order:
N =log10
([√1δ22− 1
]/ε
)log10
(Ωs
Ωp
)δ2 is the maximum allowed value for the amplitude function at the stop band edge:
δ2 = |H(Ωs)|max
The denominator of the system function is a Butterworth polynomial if Ωp = 1.These polynomials can be found in Table 2.1. For a general Ωp
H(s) =K(
sΩp
)N+ aN−1
(s
Ωp
)N−1+ · · · + a1
(s
Ωp
)+ 1
where a1, . . . , aN−1 can be found in Table 4.1.
23
Table 4.1
Coefficients aν in Butterworth polynomials sN + aN−1sN−1 + . . . + a1s + 1
N a1 a2 a3 a4 a5 a6 a7
1
2√
23 2 24 2.613 3.414 2.6135 3.236 5.236 5.236 3.2366 3.864 7.464 9.141 7.464 3.8647 4.494 10.103 14.606 14.606 10.103 4.4948 5.126 13.138 21.848 25.691 21.848 13.138 5.126
Table 4.2
Factorized Butterworth polynomials for Ωp = 1. For Ωp = 1 let s → s/Ωp.
N1 (s + 1)
2 (s2 +√
2s + 1)3 (s2 + s + 1)(s + 1)4 (s2 + 0.76536s + 1)(s2 + 1.84776s + 1)5 (s + 1)(s2 + 0.6180s + 1)(s2 + 1.6180s + 1)
6 (s2 + 0.5176s + 1)(s2 +√
2s + 1)(s2 + 1.9318s + 1)7 (s + 1)(s2 + 0.4450s + 1)(s2 + 1.2465s + 1)(s2 + 1.8022s + 1)8 (s2 + 0.3896s + 1)(s2 + 1.1110s + 1)(s2 + 1.6630s + 1)(s2 + 1.9622s + 1)
3.1.2 Chebyshev Filters
|H(Ω)| =K√
1 + ε2T 2N
(ΩΩp
)
|H( )|
p
2K
1+
RippleK
The amplitude function always passes this point.
N odd
N even
Not necessary the 3 dB frequency
Ripple = 10 · log(1 + ε2) dB.K = The maximum value of the amplitude function.
24
K = the value of the amplitude function for Ω = 0 when N is even.TN( Ω
Ωp) is a Chebyshev polynomial. (Also denoted CN
(ΩΩp
)). These can be found
in Table 4.3 for Ωp = 1. For Ωp = 1 let Ω → ΩΩp
in Table 4.3.
System function
H(s) =
K · a0 ·
1 N odd1√
1+ε2 N even(s
Ωp
)N+ aN−1
(s
Ωp
)N−1+ · · · + a0
where ε, a0, . . . , aN−1 can be found in Table 4.4.The locations of the poles for H(s) can be found in Table 4.5 for Ωp = 1. ForΩp = 1 the pole locations are multiplied with Ωp.
Table 4.3
Chebyshev polynomials.
TN(Ω) =
cos(N arccos Ω) |Ω| ≤ 1
cosh(N arccoshΩ) |Ω| ≥ 1Ω = 2πF
or
TN(Ω) =
(Ω +
√Ω2 − 1
)N+
(Ω +
√Ω2 − 1
)−N
2|Ω| ≥ 1
Recursive calculation
TN+1(Ω) = 2ΩTN(Ω) − TN−1(Ω)
Filter order:
N =arccosh
([√1δ22− 1
]/ε
)arccosh
(Ωs
Ωp
)δ2 is the maximum allowed stop band ripple.
Alternative for arccosh:
arccosh(x) = ln(x +
√x2 − 1
)
25
N TN(Ω)0 11 Ω2 2Ω2 − 13 4Ω3 − 3Ω4 8Ω4 − 8Ω2 + 15 16Ω5 − 20Ω3 + 5Ω6 32Ω6 − 48Ω4 + 18Ω2 − 17 64Ω7 − 112Ω5 + 56Ω3 − 7Ω8 128Ω8 − 256Ω6 + 160Ω4 − 32Ω2 + 19 256Ω9 − 576Ω7 + 432Ω5 − 120Ω3 + 9Ω
10 512Ω10 − 1280Ω8 + 1120Ω6 − 400Ω4 + 50Ω2 − 1
26
Table 4.4. Chebyshev filter coefficients aν.
0.5dB ripple (ε = 0.349, ε2 = 0.122).N a7 a6 a5 a4 a3 a2 a1 a0
1 2.8632 1.426 1.5163 1.253 1.535 0.7164 1.197 1.717 1.025 0.3795 1.172 1.937 1.309 0.752 0.1796 1.159 2.172 1.589 1.172 0.432 0.0957 1.151 2.413 1.869 1.648 0.756 0.282 0.0458 1.146 2.657 2.149 2.184 1.148 0.573 0.152 0.024
1-dB ripple (ε = 0.509, ε2 = 0.259).N a7 a6 a5 a4 a3 a2 a1 a0
1 1.9652 1.098 1.1023 0.989 1.238 0.4914 0.953 1.454 0.743 0.2765 0.937 1.689 0.974 0.580 0.1236 0.928 1.931 1.202 0.939 0.307 0.0697 0.923 2.176 1.429 1.357 0.549 0.214 0.0318 0.920 2.423 1.655 1.837 0.447 0.448 0.107 0.017
2-dB ripple (ε = 0.765, ε2 = 0.585).N a7 a6 a5 a4 a3 a2 a1 a0
1 1.3072 0.804 0.8233 0.738 1.022 0.3274 0.716 1.256 0.517 0.2065 0.705 1.499 0.693 0.459 0.0826 0.701 1.745 0.867 0.771 0.210 0.0517 0.698 1.994 1.039 1.144 0.383 0.166 0.0208 0.696 2.242 1.212 1.579 0.598 0.359 0.073 0.013
3-dB∗) ripple (ε = 0.998, ε2 = 0.995).N a7 a6 a5 a4 a3 a2 a1 a0
1 1.0022 0.645 0.7083 0.597 0.928 0.2514 0.581 1.169 0.405 0.1775 0.575 1.415 0.549 0.408 0.0636 0.571 1.663 0.691 0.699 0.163 0.0447 0.568 1.911 0.831 1.052 0.300 0.146 0.0168 0.567 2.161 0.972 1.467 0.472 0.321 0.056 0.011
*) The table was calculated with “exactly” 3dB, not with 20 · log√
2 ≈ 3.01dB.Hence ε = 1 and a0 = 1 for N = 1.
27
Table 4.5. Pole locations for Chebyshev filters.
0.5dB ripple (ε = 0.349, ε2 = 0.122).N = 1 2 3 4 5 6 7 8-2.863 -0.713 -0.626 -0.175 -0.362 -0.078 -0.256 -0.044
±j1.004 ±j1.016 ±j1.008 ±j1.005-0.313 -0.423 -0.112 -0.212 -0.057 -0.124
±j1.022 ±j0.421 ±j1.011 ±j0.738 ±j1.006 ±j0.852-0.293 -0.290 ±0.160 -0.186
±j0.625 ±j0.270 ±j0.807 ±j0.570-0.231 -0.220
±j0.448 ±j0.200
1-dB ripple (ε = 0.509, ε2 = 0.259).N = 1 2 3 4 5 6 7 8-1.965 -0.549 -0.494 -0.139 -0.289 -0.062 -0.205 -0.035
±j0.895 ±j0.983 ±j0.993 ±j0.996-0.247 -0.337 -0.089 -0.170 -0.046 -0.100
±j0.966 ±j0.407 ±j0.990 ±j0.727 ±j0.995 ±j0.845-0.234 -0.232 -0.128 -0.149
±j0.612 ±j0.266 ±j0.798 ±j0.564-0.185 -0.176
±j0.443 ±j0.198
2-dB ripple (ε = 0.765, ε2 = 0.585).N = 1 2 3 4 5 6 7 8-1.307 -0.402 -0.369 -0.105 -0.218 -0.047 -0.155 -0.026
±j0.813 ±j0.958 ±j0.982 ±j0.990-0.184 -0.253 -0.067 -0.128 -0.034 -0.075
±j0.923 ±0.397 ±j0.973 ±0.719 ±j0.987 ±j0.839-0.177 -0.175 -0.097 -0.113
±j0.602 ±j0.263 ±j0.791 ±j0.561-0.140 -0.133
±j0.439 ±j0.197
3-dB∗) ripple (ε = 0.998, ε2 = 0.995).N = 1 2 3 4 5 6 7 8-1.002 -0.322 -0.299 -0.085 -0.177 -0.038 -0.126 -0.021
±j0.777 ±j0.946 ±j0.976 ±0.987-0.1493 -0.206 -0.055 -0.104 -0.028 -0.061±j0.904 ±j0.392 ±j0.966 ±0.715 ±j0.983 ±j0.836
-0.144 -0.143 -0.079 -0.092±j0.597 ±j0.262 ±j0.789 ±j0.559
-0.114 -0.108±j0.437 ±j0.196
*) See note in Table 4.4.
28
3.2 Frequency Transformations of Analogue Filter
1. Start out from the frequencies in the filter specification in the analogue highpass-, band pass- or band stop filter.
2. Transform to the frequencies of the LP filter Ω′2 and Ω
′3.
3. Find the coefficients of the LP filter coefficients.
4. Transform back to the original filter (HP, BP, BS) by replacing s i the lowpass filter H(s); see below.
4Ω3Ω Ω l Ω2Ω2Ω3
Α(Ω) Α(Ω) Α(Ω)
ΩΩ Ω
LP HP BP BSΑ(Ω)
ΩΩ 2 Ω 3 4Ω3Ω Ω l Ω2
HP ⇒ LP Ω′2 = 1
Ω2Ω
′3 = 1
Ω3
BP ⇒ LP Ω′2 = Ω2 − Ω1 Ω
′3 = Ω4 − Ω3 Ω1Ω2 = Ω3Ω4 = Ω2
I
BS ⇒ LP Ω′2 =
Ω2I
Ω4−Ω3Ω
′3 =
Ω2I
Ω2−Ω1Ω1Ω2 = Ω3Ω4 = Ω2
I
LP HP BP BS
s → 1s s +
Ω2I
sΩ2
I
s +Ω2
I
s
29
4 Discrete Time Filters
4.1 FIR Filters and IIR Filters
H(z) = b0 + b1z−1 + . . . + bM−1z
−M+1
h(n) =
bn 0 ≤ n ≤ M − 10 Else
Linear phase FIR Filter :Symmetrical impulse response h(n) = h(M − 1 − n)Anti-symmetrical impulse response h(n) = −h(M − 1 − n)
IIR Filter
H(z) =b0 + b1z
−1 + . . . + bMz−M
1 + a1z−1 + . . . + aNz−N
h(n) = Z−1H(z)
4.2 Construction of FIR Filter
4.2.1 FIR Filter using the window method
Impulse responseh(n) = hd(n) · w(n)
with desired impulse response hd(n) and time window w(n).Filters designed using the window method have linear phase.
Desired impulse response hd(n) defined for 0 ≤ n ≤ M − 1Odd filter length
Low pass:
hd(n) =ωc
π
sin ωc
(n − M−1
2
)ωc
(n − M−1
2
)hd(
M − 1
2) =
ωc
π
Hd(ω) =
e−jω (M−1)/2 0 ≤ |ω| < ωc
0 Else
High pass:
hd(n) = δ(n − M − 1
2
)− ωc
π
sin ωc
(n − M−1
2
)ωc
(n − M−1
2
)Hd(ω) =
e−jω (M−1)/2 ωc < |ω| < π0 Else
30
Band pass:
hd(n) = 2 cos(ω0
(n − M − 1
2
))· ωc
π
sin ωc
(n − M−1
2
)ωc
(n − M−1
2
)Hd(ω) =
e−jω (M−1)/2 ω0 − ωc < |ω| < ω0 + ωc
0 Else
Band stop:
hd(n) = δ(n − M − 1
2
)− 2 cos
(ω0
(n − M − 1
2
))· ωc
π
sin ωc
(n − M−1
2
)ωc
(n − M−1
2
)Hd(ω) =
e−jω (M−1)/2 0 < |ω| < ω0 − ωc and ω0 + ωc < |ω| < π0 Else
The spectrum of the filter H(ω) = Hd(ω) ∗ W (ω).
At the cut-off frequency ωc the attenuation is 6dB.When dimensioning filters, the tables below offers a rough estimation of the requiredfilter length M .
Table 5.1
The main and sidelobe size for some common window functions.
Approximate Largest sidelobeWindow main lobe width relative main lobe
(rad) (dB)Rektangular 4π/M -13Bartlett 8π/M -27Hanning 8π/M -32Hamming 8π/M -43Blackman 12π/M -58
Table 5.2
Sidelobe size for some filters designed using the window method.
ApproximateWindow largest sidelobe
(dB)Rektangulr -20Bartlett -27Hanning -40Hamming -50Blackman -70Kaiser (β = 4.54) -50Kaiser (β = 6.76) -70Kaiser (β = 8.96) -90
31
Window functions defined for 0 ≤ n ≤ M − 1 (Odd filter length M)
Rectangular windoww(n) = 1
Bartlett (Triangular window)
w(n) = 1 −∣∣∣n − M−1
2
∣∣∣M−1
2
Hanning window
w(n) = 0.5
1 + cos
2π(n − M−1
2
)M − 1
=
= 0.5(1 − cos
2πn
M − 1
)
Hamming window
w(n) = 0.54 + 0.46 cos2π
(n − M−1
2
)M − 1
=
= 0.54 − 0.46 cos2πn
M − 1
Blackman window
w(n) = 0.42 + 0.5 cos2π
(n − M−1
2
)M − 1
+
+0.08 cos4π
(n − M−1
2
)M − 1
=
= 0.42 − 0.5 cos2πn
M − 1+ 0.08 cos
4πn
M − 1
Kaiser window
w(n) =I0
(β√
1 − [2(n − M−12
)/(M − 1)]2)
I0(β)
I0(x) is a Bessel function, whose value can be calculated according to
I0(x) = 1 +L∑
k=1
[(x/2)k
k!
]2
Usually L < 25.The value of β is decided from the stop band attenuation D.
32
Stop band attenuation D βD ≤ 21dB 0
21dB < D < 50dB 0, 5842(D − 21)0,4 + 0, 07886(D − 21)D ≥ 50dB 0, 1102(D − 8, 7)
Filter length M :
M ≥ D − 7, 95
14, 36∆f
∆f = fs − fp
fp Pass band frequency (normalized frequency)fs Stop band frequency (normalized frequency)
33
4.2.2 Equiripple FIR Filter
Dimensioning of equiripple filters when using the Remez algorithm. Approximatelyaccording to Kaiser.
ds
fs
1+dp
p1-
pf
f
|H(f)|
1d
Filter length M =−20 log10
(√δpδs
)− 13
14.6∆f+ 1
∆f = fs − fp
4.2.3 FIR Filters using Least-Squares
MinimizingE =
∑n
[x(n) ∗ h(n) − d(n)]2
yieldsM−1∑n=0
h(n)rxx(n − ) = rdx() = 0, . . . ,M − 1
and
Emin = rdd(0) −M−1∑k=0
h(k)rdx(k)
where rxx() is the correlation function for x(n) and rdx() is the cross correlationbetween d(n) and x(n).This can be written on matrix form as
Rxx · h = rdx
h = R−1xx · rdx
Emin = rdd(0) − hT · rdx
34
4.3 Constructing IIR Filters starting fromAnalogue Filters
4.3.1 The Impulse-Invariant Method
h(n) = ha(t)|t=nT = ha(nT )
H(z) =∞∑
n=0
ha(nT )z−n
1.
ha(t) = e−σ0t ←→ Ha(s) =1
s + σ0
⇒ H(z) =1
1 − e−σ0T z−1
2.
ha(t) = e−σ0t cos Ω0t ←→ Ha(s) =s + σ0
(s + σ0)2 + Ω20
⇒ H(z) =1 − z−1e−σ0T cos Ω0T
1 − 2z−1e−σ0T cos Ω0T + z−2e−2σ0T
3.
ha(t) = e−σ0t sin Ω0t ←→ Ha(s) =Ω0
(s + σ0)2 + Ω20
⇒ H(z) =z−1e−σ0T sin Ω0T
1 − 2z−1e−σ0T cos Ω0T + z−2e−2σ0T
4.3.2 Bilinear Transformation
Frequency transformation (“prewarp”)
Ωprewarp =2
Ttan
ω
2
Analogue construction of filters in the variable Ωprewarp = 2πFprewarp ( ω = 2πf).
H(z) = Ha(s) where s =2
T
1 − z−1
1 + z−1
T is a normalization factor (can often be set to 1).
35
4.3.3 Quantizing of Coefficients
Change in pole locations when the coefficients a1, . . . , ak is changed ∆a1, . . . , ∆ak
∆pi ≈ ∂pi
∂a1
∆a1 + · · · + ∂pi
∂ak
∆ak
If using normal form (Direct form II) then
∂pi
∂aj
=−pk−j
i
(pi − p1)(pi − p2) . . . (pi − pk)︸ ︷︷ ︸k−1 factors
(pi − pi) should not be included
4.4 Lattice Filters
z -1
+
+g (n)0 g (n)1
z -1
f (n)0 f (n)1
+
+f (n)2
z -1 +
+
K 1 K 2
K 1 K 2
f (n) = y(n)M-1
g (n)2 M -1g (n)
. . .
. . .
Kx(n)
Lattice FIR
z -1
+
+ z -1
+
+ z -1
Nf (n)
f (n)N-1 f (n)2 f (n)1
K 1
K 1
K 2
K 2
KN
KN
+
+g (n)N
+ +
g (n)0
0vv1Nv
f (n)0
g (n)2 g (n)1
v2
+
z -1
+
+
+
+ z -1
+
+ z -1
Nf (n)
f (n)N-1 f (n)2 f (n)1 f (n) = y(n)0
g (n)Ng (n)2 g (n)1 g (n)0
K 1
K 1
K 2
K 2
KN
KN
Lattice all-pole IIR
. . .
. . .
x(n)
y(n)
- - -
. . .
Lattice-ladder
. . .
. . .
x(n)
- --
36
Conversion from Lattice to Direct form:
A0(z) = B0(z) = 1
Am(z) = Am−1(z) + Kmz−1Bm−1(z)Bm(z) = KmAm−1(z) + z−1Bm−1(z)
m = 1, 2, . . . , M − 1
Relationship between Am(z) and Bm(z)
Am(z) = αm(0) + αm(1)z−1 + ... + αm(m − 1)z−m+1 + αm(m)z−m
Bm(z) = βm(0) + βm(1)z−1 + ... + βm(m − 1)z−m+1 + βm(m)z−m
Bm(z) = z−mAm(z−1)
βm(k) = αm(m − k)
k = 0, 1, . . . , m
Conversion from Direct form to Lattice:
Am−1(z) =1
1 − K2m
(Am(z) − KmBm(z))
m = M − 1,M − 2, . . . , 1
Reflection coefficient:Km = αm(m)
FIR Filter
H(z) = AN(z)
AM−1(z) = AN(z)
IIR Filter (all-pole)
H(z) =1
AN(z)
AM−1(z) = AN(z)
Lattice-LadderConversion from Direct form to Lattice-Ladder:
H(z) =CM(z)
AN(z)=
c0 + c1z−1 . . . cMz−M
AN(z)
Cm−1(z) = Cm(z) − vmBm(z)
vm = cm(m) m = 0, 1, . . . , M
37
5 Correlation
Correlation, Cross Correlation, spectrum, cross spectrum and coherence betweeninput and output signal.
Continuous time process:
y(t) = h(t) ∗ x(t)Y (F ) = H(F ) · X(F )ryy(τ) = rhh(τ) ∗ rxx(τ)Ryy(F ) = |H(F )|2Rxx(F )ryx(τ) = h(τ) ∗ rxx(τ)Ryx(F ) = H(F ) · Rxx(F )rxx(τ) =
∫t x(t)x(t − τ)dt
ryx(τ) =∫t y(t)x(t − τ)dt
γxx(τ) = Ex(t)x(t − τ)γyx(τ) = Ey(t)x(t − τ)
Discrete time process:
y(n) = h(n) ∗ x(n)Y (f) = H(f) · X(f)ryy(n) = rhh(n) ∗ rxx(n)Ryy(f) = |H(f)|2 · Rxx(f)ryx(n) = h(n) ∗ rxx(n)Ryx(f) = H(f) · Rxx(f)rxx(n) =
∑ x()x( − n)
ryx(n) =∑
y()x( − n)γxx(n) = Ex()x( − n)γyx(n) = Ey()x( − n)
Normally distributed stochastic variables. Xi ∈ N(mi, σi)
EX1X2X3X4 = EX1X2 EX3X4 + EX1X3 EX2X4 +
+ EX1X4 EX2X3 − 2m1m2m3m4
38
6 Spectrum Estimation
Spectrum Estimation
γxx(m) = Ex(n)x(n + m) autocorrelation
Γxx(f) =∞∑
m=−∞γxxe
−j2πfm power spectrum
rxx(m) =1
N
N−m−1∑n=0
x(n)x(n + m) 0 ≤ m ≤ N − 1 auto correlation (estimate)
Pxx(f) =N−1∑
m=−N+1
rxx(m)e−j2πfm =1
N
∣∣∣∣∣N−1∑m=0
x(m)e−j2πfm
∣∣∣∣∣2
power spectrum (estimate)
Periodogram
Pxx(f) =1
N
∣∣∣∣∣N−1∑m=0
x(m)e−j2πfm
∣∣∣∣∣2
power spectrum (estimate)
Erxx(m) =
(1 − |m|
N
)γxx(m) → γxx(m) when N → ∞
var(rxx(m)) ≈ 1
N
∞∑n=−∞
[γ2xx(n) + γxx(n − m)γxx(n + m)] → 0 when N → ∞
EPxx(f) =∫ 1/2
−1/2Γxx(α)WB(f − α)dα
where WB(f) is the Fourier transform of the Bartlett window(1 − |m|
N
)
var(Pxx(f)) = Γ2xx(f)
1 +
(sin2πfN
Nsin2πf
)2 → Γ2
xx(f) when N → ∞
if x(n) Gaussian.
Periodogram using DFT:
Pxx
(k
N
)=
1
N
∣∣∣∣∣N−1∑n=0
x(n)e−j2π nkN
∣∣∣∣∣2
k = 0, . . . , N − 1
39
Bartletts method
Averaging periodograms.Divide the N sample data sequence x(n) into K blocks with M samples in eachblock. The blocks are not overlapping (N = K · M).
xi(n) = x(n + iM) i = 0, 1 . . . , K − 1 and n = 0, 1 . . . , M − 1
P ixx(f) =
1
M
∣∣∣∣∣N−1∑m=0
xi(m)e−j2πfm
∣∣∣∣∣2
i = 0, 1 . . . , K − 1
PBxx(f) =
1
K
K−1∑i=0
P ixx(f)
EPBxx(f) =
1
K
K−1∑i=0
EP ixx(f) = EP i
xx(f)
EP ixx(f) =
1
M
∫ 1/2
−1/2Γxx(α)WB(f − α)dα
where WB(f) is the Fourier Transform of the Bartlett window(1 − |m|
N
)
WB(f) =1
M
(sin πfM
sin πf
)2
var(PBxx(f)) =
1
K2
K−1∑i=0
var(P ixx(f))
var(PBxx(f)) =
1
KΓ2
xx(f)
1 +
(sin2πfN
Nsin2πf
)2
Welch’s method
Averaging modified periodograms.Divide the N sample data sequence x(n) into L block with M samples in eachblock. The blocks may be overlapping. The starting point for block i is given byiD. D = M means no block overlap and D = M/2 yields 50% overlap.
xi(n) = x(n + iD) i = 0, 1 . . . , L − 1 and n = 0, 1 . . . , M − 1
40
Modified periodogram.
P ixx(f) =
1
MU
∣∣∣∣∣N−1∑m=0
xi(m)w(n)e−j2πfm
∣∣∣∣∣2
i = 0, 1 . . . , L − 1
w(n) window function
U =1
M
N−1∑n=0
w2(n)
PWxx (f) =
1
L
L−1∑i=0
P ixx(f)
EPWxx (f) =
1
L
L−1∑i=0
EP ixx(f) = EP i
xx(f)
EP ixx(f) =
∫ 1/2
−1/2Γxx(α)W (f − α)dα
W (f) =1
MU
∣∣∣∣∣M−1∑n=0
w(n)e−j2πfn
∣∣∣∣∣2
No block overlap (L = K)
var(PWxx (f)) =
1
Lvar(P i
xx(f))
≈ 1
LΓ2
xx(f)
50% block overlap (L = 2K) and Bartlett window (triangular window).
var(PWxx (f)) ≈ 9
8LΓ2
xx(f)
41
Averaging Periodograms
Quality factor
Q =[EPxx(f)]2var(Pxx(f))
Relative variance 1Q
Q ≈ time-bandwidth product.
Periodogram ∆f = 0.9M
Q = 1
Bartlett (N = K · M) ∆f = 0.9M
Q = NM
Rectangular windowNo overlap
Welch (N = L · M) ∆f = 1.28M
Q = 169· N
MTriangular window50% overlap
Welch (N = L · M) ∆f = 1.50M
Q = 31.08·2 · N
MHanning window50% overlap
Welch ∆f = 1.50M
Q = 32· N
MHanning window62,5% overlap
Blackman/Tukey ∆f = 0.6M
Q = 12· N
MRectangular window
∆f = 0.9M
Q = 32· N
MTriangular window
The resolution ∆f is calculated at the -3dB points from the main lobe of thewindow.
42
A Basic Relationships
A.1 Trigonometrical formulas
sin α = cos(α − π/2)cos α = sin(α + π/2)cos2 α + sin2 α = 1cos2 α − sin2 α = cos 2α2 sin α cos α = sin 2αsin(−α) = − sin αcos(−α) = cos αcos2 α = 1
2(1 + cos 2α)
sin(α + β) = sin α cos β + cos α sin βcos(α + β) = cos α cos β − sin α sin β2 sin α sin β = cos(α − β) − cos(α + β)2 sin α cos β = sin(α + β) + sin(α − β)2 cos α cos β = cos(α + β) + cos(α − β)
sin α + sin β = 2 sin α+β2
cos α−β2
cos α + cos β = 2 cos α+β2
cos α−β2
cos α = 12
(ejα + e−jα), sin α = 12j
(ejα − e−jα), ejα = cos α + j sin α
A cos α + B sin α =√
A2 + B2 cos(α − β)
where cos β = A√A2+B2 , sin β = B√
A2+B2
and β =
arctan BA
if A ≥ 0
arctan BA
+ π if A < 0
A cos α + B sin α =√
A2 + B2 sin(α + β)
where cos β = B√A2+B2 , sin β = A√
A2+B2
and β =
arctan AB
if B ≥ 0
arctan AB
+ π if B < 0
43
Degrees Rad sin cos tan cot
0 0 0 1 0 ±∞30 π
612
√3
21√3
√3
45 π4
1√2
1√2
1 1
60 π3
√3
212
√3 1√
390 π
2 1 0 ±∞ 0
A.2 Some Common Relationships
Sum of geometrical series
N−1∑n=0
an =
N if a = 1
1−aN
1−aif a = 1
Summation of sinusoid over an even number of periods
N−1∑n=0
ej2π kn/N =
N if k = 0,±N, . . .0 Else
A.3 Matrix Theory
Matrix notation A and vector x
A matrix A with dimension m × n and a vector x with dimension n is defined by
A =
a11 a12 . . . a1n
a21 a22 . . . a2n...
......
am1 am2 . . . amn
x =
x1
x2...
xn
The matrix A is symmetrical if aij = aji ∀ ij.I denotes the unity matrix.
Transposing matrix A
B = AT where bij = aji
(AB)T = BTAT
44
Matrix A determinant
detA = |A| =
∣∣∣∣∣∣∣∣∣∣
a11 a12 . . . a1n
a21 a22 . . . a2n...
......
am1 am2 . . . amn
∣∣∣∣∣∣∣∣∣∣=
n∑i=1
aij(−1)i+j detMij
where Mij is the resulting matrix if row i and column j in matrix A is deleted.
detAB = detA · detB
Especially, for a 2×2 matrix:
detA = a11a22 − a12a21
Inverse of matrix A
A−1A = AA−1 = I (if detA#0)
A−1 =1
detA· CT
where C is defined bycij = (−1)i+j · detMij
(AB)−1 = B−1A−1
Especially for a 2×2 matrix:
A−1 =1
detA
(a22 −a12
−a21 a11
)
Eigenvalues and Eigenvectors
The eigenvalues (λi, i = 1, 2, . . . , n) and the eigenvectors (qi, i = 1, 2, . . . , n) aresolutions for the equation system
Aq = λq or (A − λI)q = 0
The eigenvalues can be calculated as solutions to the characteristic equation (sec-ular equation) for A
det(λI − A) = λn + αn−1λn−1 + · · · + α0 = 0
det(λI−A) is called the characteristic polynomial (the secular polynomial) for A.
45
B Transforms
B.1 The Laplace Transform
B.1.1 The Laplace transform of causal signals
In the table below f(t) = 0 for t < 0 (i.e. f(t) · u(t) = f(t)).
1. f(t) = ←→ F(s) =1
2πj
∫ σ+j∞σ−j∞ F(s)estds
∫∞0− f(t)e−stdt
2.∑
ν aνfν(t) ←→ ∑ν aνFν(s) Linearity
3. f(at) ←→ 1aF( s
a) Scaling
4. 1a
f( ta) ←→ F(as) a > 0 Scaling
5. f(t − t0); t ≥ t0 ←→ F(s) e−st0 Time shift
6. f(t) · e−at ←→ F(s + a) Frequency shift
7. dnfdtn
←→ snF(s) Derivation
8.∫ t0− f(τ)dτ ←→ 1
sF(s) Integration
9. (−t)nf(t) ←→ dnF(s)dsn Derivation in
frequency domain
10. f(t)t
←→ ∫∞s F(z)dz Integration in
frequency domain
11. limt→0 f(t) = lims→∞ s · F(s) Initial valuetheorem
12. limt→∞ f(t) = lims→0 s · F(s) End value theorem
13. f1(t) ∗ f2(t) = ←→ F1(s) · F2(s)∫ t0 f1(τ)f2(t − τ)dτ = Convolution in time domain∫ t0 f1(t − τ)f2(τ)dτ
14. f1(t) · f2(t) ←→ 12πj
F1(s) ∗ F2(s) =1
2πj
∫ σ+j∞σ−j∞ F1(z) · F2(s − z) · dz
Convolution in frequency domain
15.∫∞0− f1(t) · f2(t)dt =1
2πj
∫ σ+j∞σ−j∞ F1(s) · F2(−s)ds Parseval’s relation
46
16. δ(t) ←→ 1
17. δn(t) ←→ sn
18. 1 ←→ 1s
19. 1n! tn ←→ 1
sn+1
20. e−σ0t ←→ 1s + σ0
21. 1(n − 1)!
tn−1e−σ0t ←→ 1(s + σ0)
n
22. sin Ω0t ←→ Ω0
s2 + Ω20
23. cos Ω0t ←→ ss2 + Ω2
0
24. t · sin Ω0t ←→ 2Ω0s(s2 + Ω2
0)2
25. t · cos Ω0t ←→ s2 − Ω20
(s2 + Ω20)
2
26. e−σ0t sin Ω0t ←→ Ω0
(s + σ0)2 + Ω2
0
27. e−σ0t cos Ω0t ←→ s + σ0
(s + σ0)2 + Ω2
0
28. e−σ0t sin(Ω0t + φ) ←→ (s + σ0) sin φ + Ω0 cos φ(s + σ0)
2 + Ω20
B.1.2 One-sided Laplace transform of non-causal signals
Notation
F+(s) =∫∞0− f(t)e−stdt Single sided Laplace transform,
f(t) not necessarily causal.F(s) = F+(s) For causal signals
Taking the derivative of f(t) yields
d
dtf(t) ←→ s · F+(s) − f(0−) Single derivation
dn
dtf(t) ←→ snF+(s) − sn−1f(0−)
−sn−2f (1)(0−) − . . . f (n−1)(0−) n derivations
47
B.2 The Fourier Transform of a Continuous Time Signal
Ω = 2πF
1. w(t) = F−1W (F ) = ←→ W (F ) = Fw(t) =∫∞−∞ W (F )ej2πFtdF
∫∞−∞ w(t)e−j2πFtdt
2.∑
ν aνwν(t) ←→ ∑ν aνWν(F )
3. w∗(−t) ←→ W ∗(F )
4. W (t) ←→ w(−F )
5. w(at) ←→ 1|a| W
(Fa
)
6. w(t − t0) ←→ W (F ) · e−j2πFt0
7. w(t) · ej2πF0t ←→ W (F − F0)
8. w∗(t) ←→ W ∗(−F )
9. dnw(t)dtn
←→ (j2πF )n W (F )
10.∫ t−∞ w(τ)dτ ←→ 1
j2πFW (F ) if W (F ) = 0 for F = 0
11. −j2πt w(t) ←→ dwdF
12. w1(t) ∗ w2(t) ←→ W1(F ) · W2(F )
13. w1(t) · w2(t) ←→ W1(F ) ∗ W2(F )
14.∫∞−∞ |w(t)|2dt =
∫∞−∞ |W (F )|2dF Parseval’s relation
15.∫∞−∞ w1(t) · w2(t)dt =∫∞−∞ W1(F ) · W ∗
2 (F )dF w1(t), w2(t) real
16. δ(t) ←→ 1
17. 1 ←→ δ(F )
18. u(t) ←→ 1j2πF
+ 12
δ(F )
19. e−atu(t) ←→ 1a+jΩ
48
20. e−a|t| ←→ 2aa2+Ω2
21. ej2πF0t ←→ δ(F − F0)
22. sin 2πF0t ←→ j 12δ(F + F0) − δ(F − F0)
23. sin 2πF0t · u(t) ←→ Ω0
Ω20−Ω2 + j 1
4δ(F + F0) − δ(F − F0)
24. cos 2πF0t ←→ 12δ(F + F0) + δ(F − F0)
25. cos 2πF0t · u(t) ←→ jΩΩ2
0−Ω2 + 14δ(F + F0) + δ(F − F0)
26. 1√2πσ2
e−t2/2σ2 ←→ e−(Ωσ)2/2
27. e−at sin 2πF0t · u(t) ←→ Ω0
(jΩ + a)2 + (Ω0)2
28. e−a|t| sin 2πF0|t| ←→ 2Ω0(Ω20 + a2 − Ω2)
(Ω2 + a2 − Ω20)
2 + 4a2Ω20
29. e−at cos 2πF0t · u(t) ←→ jΩ + a(jΩ + a)2 + (Ω0)
2
30. e−a|t| cos 2πF0t ←→ 2a(Ω20 + a2 + Ω2)
(Ω2 + a2 − Ω20)
2 + 4a2Ω20
31. rect(at) =
1 for |t| < 1
2a
0 Else←→ 1
asinc(F
a) a > 0
32. sinc(at) = sin(πat)πat
←→ 1a
rect(Fa) a > 0
33. repT (w(t)) =∑∞
m=−∞ w(t − mT ) ←→ 1|T | comb1/T (W (F ))
34. |T |combT (w(t)) = ←→ rep1/T (W (F ))|T |∑∞
m=−∞ w(mT )δ(t − mT )
35.∑∞
n=−∞ cnδ(t − nT ) ←→ ∑∞n=−∞
1T
cnδ(F − nT) =
∑cne
−j2πnTF
49
B.3 The Z-transform
B.3.1 The Z-transform of causal signals
1. X (z) = Z[x(n)] =∑∞
n=−∞ x(n)z−n Transform
2. x(n) = Z−1[X (z)] = 12πj
∫Γ X (z)zn−1dz Invers transform
3.∑
ν aνxν(n) ←→ ∑ν aνXν(z) Linearity
4. x(n − n0) ←→ z−n0X (z) Shift (n0 positive ornegative integer)
5. nx(n) ←→ −z ddz
X (z) Multiplication with n
6. anx(n) ←→ X(
za
)Scaling
7. x(−n) ←→ X(
1z
)Reflection of the time sequence
8.[∑n
=−∞ x()]←→ z
z−1X (z) Summation
9. x ∗ y ←→ X (z) · Y(z) Convolution
10. x(n) · y(n) ←→ 12πj
∫Γ Y(ξ)X
(zξ
)ξ−1dξ Product
11. x(0) = limz→∞X (z) (if the limit value exist) Initial value theorem
12. limn→∞ x(n) = limz→1(z − 1)X (z) End value theorem(if the unit cirle is included in the ROC)
13.∑∞
=−∞ x()y() = 12πj
∫Γ x(z)y
(1z
)z−1dz Parseval’s teorem for
real time sequences
14.∑∞
=−∞ x2() = 12πj
∫Γ X (z)X (z−1)z−1dz – ” –
50
Sequence ←→ Transform
x(n) ←→ X (z)
15. δ(n) ←→ 1
16. u(n) ←→ 11 − z−1
17. nu(n) ←→ z−1
(1 − z−1)2
18. αnu(n) ←→ 11 − αz−1
19. (n + 1)αnu(n) ←→ 1(1 − αz−1)2
20.(n + 1)(n + 2) . . . (n + r − 1)
(r − 1)!αnu(n) ←→ 1
(1 − αz−1)r
21. αn cos βn u(n) ←→ 1 − z−1α cos β1 − z−12α cos β + α2z−2
22. αn sin βn u(n) ←→ z−1α sin β1 − z−12α cos β + α2z−2
23. Fnu(n) ←→ (I − z−1F)−1
B.3.2 Single-Sided Z-transform of non-causal signals
Notation
X+(z) =∑∞
n=0 x(n)z−n Single-sided z-transform, x(n) notnecessarily causal.
X (z) = X+(z) For causal signals
Shifting x(n)yields:i) one step shift
x(n − 1) ←→ z−1X+(z) + x(−1)
x(n + 1) ←→ zX+(z) − x(0) · zii) n0 step shift (n0 ≥ 0)
x(n − n0) ←→ z−n0X+(z) + x(−1)z−n0+1 +
+x(−2)z−n0+2 + . . . + x(−n0)
x(n + n0) ←→ zn0X+(z) − x(0)zn0 − x(1)zn0−1 − . . . − x(n0 − 1)z
51
B.4 Fourier Transform for Discrete Time Signal
1. X(f) = F(x(n)) = Transform=
∑∞=−∞ x()e−j2πf ω = 2πf
2. x(n) =∫ 1/2−1/2 X(f)ej2πfndf = Inverse transform
= 12π
∫ π−π X(f)ejωndω
3.∑
aνxν(n) ←→ ∑ν aνXν(f) Linearity
4. x(n − n0) ←→ X(f) · e−j2πfn0 Shift
5. x(n)ej2πf0n ←→ X(f − f0) Frequency translation
6. x(n) · cos 2πf0n ←→ 1
2[X(f − f0) + X(f + f0)]
Modulation
7. x(n) · sin 2πf0n ←→ 1
2j[X(f − f0) − X(f + f0)]
Modulation
8. x ∗ y ←→ X(f) · Y (f) Convolution
9. x · y ←→ ∫ 1/2−1/2 X(λ) · Y (f − λ)dλ Product
10.∑∞
=−∞ x()y() = Parseval’s teorem
=∫ 1/2−1/2 X(f)Y ∗(f)df for real time sequences
11. X(f) = X (ejω) if x(n) = 0 for n < n0 and∑∞=−∞ |x()|2 < ∞
(Valid for for example: 18,19,20,21 och 22in the Z-transform table for |α| < 1)
12. δ(n) ←→ 1
13. δ(n − n0) ←→ e−jωn0
14. 1 ∀n ←→ ∑∞p=−∞ δ(f − p)
15. u(n) ←→ 1
2
∑∞p=−∞ δ(f − p) +
1
2+ 1
j · 2 · tan(πf)
52
16. 2f1 · sinc(2f1 · n) = 2f1sin(2πf1n)
2πf1n
←→ rectp
(f
2f1
)=
1 |f − n| < f1 < 1/2 , n integer
0 ElseIdeal LP filter
17. 4f1sinc(2f1n) cos(2πf0n)
←→ rectp
(f − f0
2f1
)+ rectp
(f + f0
2f1
)Ideal BP Filter
18. 2πf1n cos 2πf1n − sin 2πf1nπn2
←→ (j2πf)p =
j2π(f − n) |f − n| < f1 < 1/2 , n integer
0 Else“Derivating” system
19. cos(2πf0n) ←→ 1
2
∑∞p=−∞[δ(f − f0 − p) + δ(f + f0 − p)]
20. α|n| ←→ 1 − α2
1 + α2 − 2α cos 2πf
21. α|n| cos(2πf0n)
←→ 1 − α2
2
[1
1 + α2 − 2α cos 2π(f + f0)+ 1
1 + α2 − 2α cos 2π(f − f0)
]
22. pr(n) =
1 |n| ≤ M − 12
0 Else
M odd
←→ Pr(f) =sin(πfM)sin(πf)
Rectangular window
53
B.5 Some DFT Properties
Time Frequencyx(n), y(n) X(k), Y (k)x(n) = x(n + N) X(k) = X(k + N)x(N − 1) X(N − k)x((n − 1))<N> X(k)e−j2πk1/N
x(n)ej2π1n/N X((k − 1))<N>
x∗(n) X∗(N − k)x1(n) ©∗ x2(n) X1(k)X2(k)x(n) ©∗ y∗(−n) X(k)Y ∗(k)x1(n)x2(n) 1
NX1(k) N X2(k)∑N−1
n=0 x(n)y∗(n) 1N
∑N−1k=0 X(k)Y ∗(k)
54
