Signals and Systems in Bioengineering (Fourier) Transforms · Signals and Systems in Bioengineering (Fourier) Transforms João Sanches ([email protected], Ext: 2195) Department

SignalsandSystemsinBioengineering

(Fourier)Transforms

JoãoSanches([email protected],Ext:2195)

DepartmentofBioengineeringInstitutoSuperiorTécnico/UniversityofLisbon

4thyear,1stSemester(ECTS:6.0)

2020/2021

SignalsandSystemsinBioengineering,SSB,JoãoMiguelSanches,DBE/IST,1stSem,2020/2021

IntegralTransform

Atransformisanoperator(function)thatmapsavectorspaceXinotherspaceY(couldbethesame)

Definition:LetVbeavectorspaceoffunctions.Anintegrallinear

transformisalinearmappingasfollowsF(ω)iscalledtransformoff(t)andK(t,ω)isthekernelofthetransform.

T : X→Y

f :R→CT :V →V

T (f (t)) = f (t)K(t,ω)dt ≡ F(ω)a

b

∫

= f (t),K(t,ω)t


CommonTransforms(ofcontinuoussignals)


L(f (t),s) = F(s) = f (t)e−st dt0

∞

∫ = f (t),est

F(f (t),ω) = F(ω) = f (t)e− jωt dt−∞

∞

∫ = f (t),e jωt

F(f (t),ω) = F(ω) = f (t)cos(ωt)dt0

∞

∫ = f (t),cos(ωt)

F(f (t),ω) = F(ω) = f (t)sin(ωt)dt = f (t),sin(ωt)0

∞

∫

Laplace Transform:

Fourier Transform:

Cosine Transform:

Sine Transform:

CommonTransforms(ofdiscretesignals)


Z Transform:

TZ x(n),z( ) = X (z) = x(n)z−n−∞

∞

∑ = x,p , p(n) = zn ∈C

Fourier Transform:

FT (x(n),ω) = F(ω) = x(n)e− jωn−∞

∞

∑ = x,p , p(n) = e jωn

Cosine Transform:

CF(x(n),k) = X (k) = 2Nc(k) x(n)cos πk(2n +1)

N

⎛

⎝⎜

⎞

⎠⎟

n=0

N−1

∑

Theorem•  Anorthonormalsetoffunctions{p1,p2,…,p∞},where,isa

completesetinavectorspaceSwithinnerproductandinducednormifthefollowingequivalentconditionshold

–  Foranyx∈S,

–  Foranyε>0thereisaN<∞suchthatforanyn≥N

–  TheParsevalequalityholds:–  If<x,pi>=0foranyi,thenx=0

–  Thereisnonon-zerofunctionf∈Sforwhichtheset{p1,p2,…,p∞}∩fformanorthogonalset


x = ckpkk=1

∞

∑ , ck = x,pk

x − ckpkk=1

n

∑ < ε

x 2= ck

2

k=1

∞

∑ xyyx, Hn

kkk yx ==∑

=1

TheinnerproductinCnisdefinedas

pk ,pr = δ(k − r )

OrthogonalbasisandProjection


x = projpjxk=1

∞

∑ + e = ckpk + ek=1

∞

∑ ⇒ e = x − ckpkk=1

∞

∑

e,pi = 0⇒ x − ckpkk=1

∞

∑ ,pi = 0⇒ x,pi = ck pk ,pik=1

∞

∑

x,pi = ci pi ,pi ⇒ ci =x,pi

pi ,pi=x,pi

pi2

projpix = cipi =x,pipi

2 pi

ck =x,pipi ,pi

=x pi cos(θi )

pi2 =

x cos(θi )pi

=projpixpi

sign(cos(θ ))

Ifthebasisisorthogonal, pi ,p j = Aiδ(i − j )

x

piprojpix

Fourierdecompositiondiscretesignals


FourierTransformofdiscretesignalsEnergylimitedsignals

•  Thebasisfunctionsarecomplexexponentials

•  Kernel:

•  Thecoefficientsareobtainedwiththeinnerproduct


p(ω) = p(n,ω)⎡⎣ ⎤⎦ where p(n,ω) = e jωn

ϕ(n,ω) = e jωn

x2(n)−∞

∞

∑ <∞

c(ω) = X (ω) = x,e jωn = x (n)e− jωn

n=−∞

∞

∑

TheFouriertransformofdiscretesignalsisnotdiscrete,itiscontinuous.

ComplexExponential


e jωn = e jω(n+N) = e jωne jωN = e jωn⇒

e jωN =1⇒ωN =2kπ⇒ω =2kπ /N

•  Whatarecomplexexponentials?

•  Arecomplexexponentialsperiodic?

e jΩt ,t ∈ R

e jωn ,n∈ Z

es = eσ + jω = eσ e jω

= eσ (cos(ω)x

+ jsin(ω)y)

ωω=0

ω=π/2

ωjez =

Exercises


Figura 1: Adaptive filter.

• 2 a) h(n) = [1, 0, 0, 0].

• 2 b) h(n) = [�1, 0, 0, 0].

• ⌅ c) h(n) = [0,�1, 0, 0].

• 2 d) None

Take into account that f(n�n0) = f(n)⇤�(n�n0) and assuming that h(n) = ��(n�n0)then y(n) = x(n) ⇤ h(n) = x(n) ⇤ [��(n� n0)] = �x(n� n0) = �[�d(m+ 1)]|m=n�n0 =d(n� n0 + 1) ) y(n) = d(n) if n0 = 1 ) h(n) = ��(n� 1).

4. Consider the canonical adaptive filter displayed in Fig. 1 where d(n) = g(n)⇤x(n), x(n)is white Gaussian noise, x(n) ⇠ N (0, �2) and h(n) and g(n) are P length FIR filters(where g(n) is known). In these conditions what is the optimal impulse response of theFIR filter h(n) that minimizes the norm of the error, kek?

• ⌅ a) h(n) = g(n).

• 2 b) h(n) = g(�n).

• 2 c) h(n) = g⇤(n).

• 2 d) None

where ⇤ denotes conjugation.

If h(n) = g(n) ) y(n) = g(n)⇤x(n) ) e(n) = d(n)�y(n) = g(n)⇤x(n)�g(n)⇤x(n) = 0

5. What is the period of the signal x(n) = exp(2⇡(j � 1)n/10)?

• 2 a) 10/(j � 1) samples

• 2 b) 10/p2 sample

• 2 c) 10 sample

• ⌅ d) None

2The signal x(n) = exp(2⇡(j � 1)n/10) = exp(j2⇡n/10)exp(�⇡n/5) is not periodicbecause the second term, exp(�⇡n/5), is a real decaying exponential.

6. The Fast Fourier Transform (FFT) optimizes the computation of the DFT by removingcompletely redundant computations. The core of the FFT algorithm, called butterfly,is a structure that computes a 2-length DFT vector of Fourier coe�cients DFT2(x) =X = [X(0), X(1)]T from 2 length sequences, x = [x(0), x(1)]T . Using matrix notation

X = Wx

where W is one of the following 2⇥ 2 square matrices. What is that matrix?

• 2 a) W = [1, 0; 0, 1].

• ⌅ b) W = [1, 1; 1,�1].

• 2 c) W = [1, 0; 1, 0].

• 2 d) None

X(k) =PN�1

k=0 x(n)e�2⇡N kn

.

For N = 2 )X(0) = x(0)e�pi⇤0⇤0 + x(1)e�pi⇤0⇤1 = x(0) + x(1)X(1) = x(0)e�pi⇤1⇤0 + x(1)e�pi⇤1⇤1 = x(0)� x(1) )X = Wx where X = [X(0), X(1)]T ,x = [x(0), x(1)]T and W = [1, 1; 1� 1]

7. Consider the chirp signal x(t) = sin(2⇡f(t)t) with 0 t 1 seconds and f(t) =100 + 900t2 Hz. What is the maximum frequency of the spectrum of x(t)?

• 2 a) 2900 Hz

• 2 b) 1800 Hz

• 2 c) 1000 Hz

• ⌅ d) None

x(t) = sin(�(t)) where �(t) = 2⇡f(t)t!(t) = d�(t)/dt = 2⇡finst(t) = 2⇡[f(t) + tf(t)] where f(t) = 1800t )finst(t) = 100 + 900t2 + 1800t2 = 100 + 2700t2, a monotonic increasing function withtime, t ) fmax = finst(1) = 100 + 2700 = 2800 Hz

8. Consider the analog signal x(t) = sin(2⇡f1t) + sin(2⇡f2t) with f1 = 1000 Hz andf2 = 1010 Hz. What are the appropriated sampling rate, fs, and FFT length, N , thatshould be used to discriminate both peaks in the spectrum of the discrete (sampled)version of x(t), xd(n) = x(nTs)?

3

4. What is the frequency of the discrete signal x(n) = exp(j2n/7)?

• 2 a) 2/7.

• 2 b) 2/14.

• 2 c) 2⇡/14.

• 2 d) None

5. Consider the Linear Time Invariant (LTI) filter with the following transfer function

H(z) =1� 0.1z

�1

1� 0.7z�1 + 0.1z�2(1)

What is the corresponding time recursion that can be used to implement the filter?

• 2 a) y(n) = x(n)� 0.1x(n� 1) + 0.7y(n� 1)� 0.1y(n� 2).

• 2 b) y(n) = x(n)� 0.1x(n� 1)� 0.7y(n� 1) + 0.1y(n� 2).

• 2 c) y(n) = x(n) + 0.1x(n� 1) + 0.7y(n� 1)� 0.1y(n� 2).

• 2 d) None

6. Consider a 10 length signal x = [0; 1; 2; 3; 4; 5; 6; 7; 8; 9]. Sample the Fourier transform

of x, X(!), at 8 evenly spaced frequencies, X8(k), and compute y(n) = DFT�18 (X), for

n = [0, 1, ..., 7], where DFT�18 () denotes a 8 length DFT inversion operator.

What is y(n) ?

• 2 a) y(n) = [0; 1; 2; 3; 4; 5; 6; 7].

• 2 b) y(n) = [8; 9; 2; 3; 4; 5; 6; 7].

• 2 c) y(n) = [8; 10; 2; 3; 4; 5; 6; 7].

• 2 d) None

7. Consider the following transfer function of a filter:

H(z) =1� 0.5z

�1

1� (3/2)z�1 + (13/16)z�2(2)

with poles p1,2 =34 ± j

12 . What is central frequency of this filter?

• 2 a) !0 = 0 rad/sample.

• 2 b) !0 = 1 rad/sample.

• 2 c) !0 = arctan(2/3).

2

FourierTransformofdiscretetimelimitedsequences

•  Periodicity(ShowtheperiodicityofX(ω))


Analysis X (ω) = x (n)e− jωn

n=−∞

∞

∑

Synthesis x (n) = 12π

X (ω)e jωn dω−π

π

∫

%

&

''

(

''

X (ω + 2π ) = x (n)e− j (ω+2π )n

n=−∞

∞

∑ = x (n)e− jωn e− j 2πn

1!

n=−∞

∞

∑ = x (n)e− jωn

n=−∞

∞

∑

= X (ω)

2π

X (ω)

4π−2π

ω

0

n

x(n)

x2(n)−∞

∞

∑ <∞

FourierSeriesofperiodicsignals-Nonlimitedenergy

•  LetbeaperiodicsignalofperiodNandafrequencyof•  Aperiodicsignalcanbeexpressedaslinearcombinationof

periodicbasisfunctions

whereareperiodicfunctionswithmultiplefrequencyof,


!x (n) = ck !ϕk (n)k∑

!ϕk (n)

ω0

x(n)ω0 =2π /N

x2(n)−∞

∞

∑ =∞

ωk = kω0 = 2kπ /N

n

x(n)

FourierSeriesPeriodicsequences

•  Considerthefollowingperiodicbasisfunctions

withfrequencyrad/sample•  Itisperiodicinnandk.

•  ThismeansthatonlyNfunctionsaredifferent,e.g.,

•  ThissetofNperiodicfunctionsareabletogenerateanyperiodicdiscretesequenceofperiodNwithlimitedenergy.


ϕk (n) = ej 2πNkn

ωk =2πk /N

Analysis X (k ) = !x (n)e− j 2π

Nkn

n=0

N−1

∑

Synthesis !x (n) = 1N

X (k )ej 2πNkn

k=0

N−1

∑

#

$

%%

&

%%

Exercise:DeriveX(k)=<x,φk(n)>

ϕ(k+N )(n) =1Nej 2πN(k+N )n

=1Nej 2πNknej 2πNNn

1!"# =

1Nej 2πNkn=ϕk (n)

ϕk (n +N ) =1Nej 2πNk (n+N )

=1Nej 2πNknej 2πNkN

1!"# =

1Nej 2πNkn=ϕk (n)

ϕk (n) k = 0,1,!,N −1

ϕN (n) =ϕ0(n), ϕN+1(n) =ϕ1(n),!,ϕN+k (n) =ϕk (n)

Exercises


• 2 d) None

4. Consider a 10 length signal x = [0; 1; 2; 3; 4; 5; 6; 7; 8; 9]. Sample the Fourier transformof x, X(!), at 8 evenly spaced frequencies, X8(k), and compute y(n) = DFT

�18 (X), for

n = [0, 1, ..., 7], where DFT�18 () denotes a 8 length DFT inversion operator.

What is y(n) ?

• 2 a) y(n) = [0; 1; 2; 3; 4; 5; 6; 7].

• 2 b) y(n) = [8; 9; 2; 3; 4; 5; 6; 7].

• 2 c) y(n) = [8; 10; 2; 3; 4; 5; 6; 7].

• 2 d) None

5. Consider a 8 length DFT, X(k) = [1;X(1); 1� j; j� j; 1+ j; 1� j], of a real signal x(n).What is the value of X(1)?

• 2 a) X(1) = 0.

• 2 b) X(1) = 1 + j.

• 2 c) X(1) = 1� j.

• 2 d) None

6. Consider a band pass filter with the following transfer function;

H(z) =1

1� (3/2)z�1 + (13/16)z�2(2)

with poles p1,2 =34 ± j

12 . What is central frequency of this filter?

• 2 a) !0 = 0 rad/sample.

• 2 b) !0 = 1 rad/sample.

• 2 c) !0 = arctan(2/3).

• 2 d) None

7. The inner product < �k(n),�r(n) > with

�⌧ (n) =1pNej 2⇡N ⌧n (3)

where k, r and ⌧ are integers and N is the total length of the signals,is

• 2 a) �(k � r).

2• 2 b) 0.

• 2 c) 1.

• 2 d) None

8. What is the period of the signal y(n) = cos(n)?

• 2 a) 1 sample.

• 2 b) 2⇡ rad/sample.

• 2 c) 1 second.

• 2 d) None

Problem (4) Consider the following model describing noisy observations

y(n) = x(n) + ⌘(n) (4)

where ⌘(n) is additive white Gaussian noise with normal distribution, ⌘(n) ⇠ N (0, �2).y = [y(0), y(1), ..., y(N � 1)]T are the noisy observations and x(n) the unknown signalto estimate.

The maximum likelihood (ML) estimation of x = [x(0), x(1), ..., x(N � 1)]T can be com-puted by minimizing the following energy function,

J = k(Ax� y)k2 (5)

where A is a known N ⇥N matrix modelling the blur e↵ect.

1. Derive the close form solution for the ML estimate of x from the observation y vector.

2. The minimization of (5) is an ill-posed problem. To regularize the solution a modifiedenergy function is proposed,

J = k(Ax� y)k2 + ↵

N�1X

n=1

(x(n)� x(n� 1))2 (6)

Derive the close form solution of the minimizer of (6).

3

FourierSeriesandTransformDiscreteSignals


Analysis X (k ) = !x (n)e− j 2π

Nkn

n=0

N−1

∑

Synthesis !x (n) = 1N

X (k )ej 2πNkn

k=0

N−1

∑

#

$

%%

&

%%

FourierSeries

Analysis X (ω) = x (n)e− jωn

n=−∞

∞

∑

Synthesis x (n) = 12π

X (ω)e jωn dω−π

π

∫

%

&

''

(

''

FourierTransform

RelationbetweenFourierSeriesandTransformFrequencysampling


X (ω)ω=ωk =

2πNk= x (n)e− jωkn

n=−∞

∞

∑ = !x (n)e− j 2π

Nkn

0

N−1

∑Compute over a period! "## $##

= X (k )

ω1 =2πNZ

0=ω

↑ω

N −1

!x(n)

nN0−N +1 −1

ωk =2πNk

N −1

x (n)

nN0−1

RelationbetweenFourierSeriesandTransformFrequencysampling


X (ω)ω=ωk =

2πNk= x (n)e− jωkn

n=−∞

∞

∑ = !x (n)e− j 2π

Nkn

0

N−1

∑Compute over a period! "## $##

= X (k )

MatrixRepresentation


Analysis: X =WNx

Synthesis: x = 1NWN

HX

where x = x(0) x(1) ... x(N −1)"#

$%T

and

X = X(0) X(1) ... X(N −1)"#

$%T

wN = e− j2π

N

DFTProperties


x (n)is real and even (par) ⇒ X (k ) is real and even (par)x (n)is real and odd (Impar) ⇒ X (k ) is imaginary and odd (impar)

x (n) = xe (n)+ xo (n)⇒ X (k )

xe (n) = x (n)+ x (−n)2

=Even(x (n))⇒ X (k ) =Re(X (k ))

xo (n) = x (n)− x (−n)2

=Odd (x (n))⇒ X (k ) = j Im(X (k ))wN = e− j 2π

N

Exercises


• 2 b) 1.

• ⌅ c) j.

• 2 d) None

5. What is the period of the signal y(n) = sin(n)?

• 2 a) 1 sample.

• 2 b) 2⇡ rad/sample.

• 2 c) 1 second.

• ⌅ d) None

6. Let g(x, y) = sin(d(x, y)) be a function of two vectors x and y, where d(x, y) is a metricfunction. The function g(x, y)

• 2 a) is a metric function because it is strictly non negative.

• 2 b) it is not a metric function because g(x, x) 6= 0

• 2 c) it is a metric function because its value increases when kx� yk increases.

• ⌅ d) None

7. Let x(n) and y(n) two discrete sequences of length 16 with DFT coe�cients X16(k)

and Y16(k) respectively, where Y (k) =

(X(k) for k even

�X(k) for k odd.

What is the right option?

• ⌅ a) y(2) = x(10).

• 2 b) y(1) = x(11).

• 2 c) y(0) = x(12).

• 2 d) None

8. The goal is to filter, in real time, an audio signal from a microphone with a 50 lengthimpulse response FIR filter. The signal should be processed with a 500 sample lengthblocks and the convolution is performed by using a 512 length FFT algorithm. Whatis the number of overlapped samples of the input blocks.

• 2 a) 12.

• ⌅ b) 37.

• 2 c) 50.

• 2 d) None

2

• 2 a) 100kHz

• 2 b) 250kHz

• 2 c) 500kHz

• 2 d) None

4. [T:2] Consider the finite length sequence x(n) = {1, 2, 3} and the sequence y(n) =x((2� n)5). What is the value of y(3).

• 2 a) 0

• 2 b) 1

• 2 c) 2

• 2 d) None

5. [T:2] Consider the complex signal x(n) = {1, 1� j, 0, 2� j, 3,�2 + j, 2j}. What is the8 length DFT value for k = 8, X8(8)?

• 2 a) 5 + j

• 2 b) 0

• 2 c) 6

• 2 d) None

6. [T:2] What is the period of the signal sin(0.01⇡n)?

• 2 a) 200

• 2 b) it is not periodic

• 2 c) 0.01

• 2 d) None

7. [T:2] The goal is to filter, in real time, an audio signal from a microphone with a 25length impulse response FIR filter. The signal should be processed with a 500 samplelength blocks and the convolution is performed by using a 512 length FFT algorithm.What is the number of overlapped samples of the input blocks?

• 2 a) 24.

• 2 b) 12.

• 2 c) 0.

• 2 d) None

2

Instituto Superior Tecnico / University of Lisbon

Departament of Bioengineering

Master on Biomedical Engineering

Signals and Systems in Bioengineering

1st Semester de 2014/2015

Joao Miguel Sanches

Test 1

Novembro 13, 2014

Name : Number:

The duration of the test is 1h30m. The score of each item is 2 when right and �0.5 if

wrong. Only one option can be selected in each question.

1. Consider the complex signal x(n) = [0; j; 1 + 3j;�1 � j; 0; 3;�2j; 1 � j]. What is the

value of X8(k) for k = 8?

• 2 a) 0.

• 2 b) 4.

• 2 c) 4� j.

• 2 d) None

2. Consider the signal x(n) = [3; 2; 1; 0; 1; 2; 3; 4]. What is the option where the 8-length

DFT is real?

• 2 a) x((n� 1)8).

• 2 b) x((n+ 1)8).

• 2 c) x((n� 2)8).

• 2 d) None

3. Consider the 4-length and 8-length sequences x4(n) and y8(n) respectively. Let also

w(n) = x(n) ⇤ y(n) and z(n) = x(n) ? y(n) where ⇤ and ? denote the linear and 8-length

circular convolutions respectively. Select the right option.

• 2 a) z(0) = w(0).

• 2 b) z(1) = w(0).

• 2 c) z(4) = w(4).

• 2 d) None

1

FastFourierTransform(FFT)•  Complexmultiplication:(a+jb)(c+

jd)=(ac-bd)+j(ad+bc)hasacomputationalcomplexityoffour(4)realmultiplications.

•  ThecomputationalcomplexityofaNdimensionDFT,usingthedefinition,is4N2realmultiplications.


•  FFThasacomputationalcomplexityO(Nlog2(N)),e.g.,N=1024⇒G=4N2/(4Nlog2(N))=102.4andifN=4096thegainisG=341:3.

•  Thecomputationalgainisobtainedbyreducingthecomputationalredundancy.

FFTcomputationalgain


Butterfly


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Signals and Systems in Bioengineering (Fourier) Transforms · Signals and Systems in Bioengineering (Fourier) Transforms João Sanches ([email protected], Ext: 2195) Department