Signals and Systems in Bioengineering (Fourier) Transforms João Sanches ([email protected], Ext: 2195) Department of Bioengineering Instituto Superior Técnico / University of Lisbon 4 th year, 1 st Semester (ECTS: 6.0) 2020/2021 Signals and Systems in Bioengineering, SSB, João Miguel Sanches, DBE/IST, 1st Sem, 2020/2021
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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
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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?
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?
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).
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
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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.