USEFUL FORMULAS Properties of valid Autocorrelation function (ACF) The ACF of a signal is defined as the measure of similarity or coherence between a signal and ts time delayed version. • ACF of a real function x(t): Rxx(T) = Ext) xx(t+T)), where T is a variable called time-shift (delay) parameter • ACF is an even function • • ACF reaches its peak at the origin: R(T) R(0) ACF of a periodic function is, itself, periodic with the same period Autocorrelation of the sum of two completely uncorrelated functions (the cross-correlation is zero for all) is the sum of the autocorrelations of each function separately • ACF is related to the power spectral density via the Fourier transform Rxx (T→ 0) = m₂(x); Rxx (T→ Infinity) = [m₂(x)]2 O Choice 1 O Choice 2 O Choice 3 O Choice 4 O Choice 5
USEFUL FORMULAS Properties of valid Autocorrelation function (ACF) The ACF of a signal is defined as the measure of similarity or coherence between a signal and ts time delayed version. • ACF of a real function x(t): Rxx(T) = Ext) xx(t+T)), where T is a variable called time-shift (delay) parameter • ACF is an even function • • ACF reaches its peak at the origin: R(T) R(0) ACF of a periodic function is, itself, periodic with the same period Autocorrelation of the sum of two completely uncorrelated functions (the cross-correlation is zero for all) is the sum of the autocorrelations of each function separately • ACF is related to the power spectral density via the Fourier transform Rxx (T→ 0) = m₂(x); Rxx (T→ Infinity) = [m₂(x)]2 O Choice 1 O Choice 2 O Choice 3 O Choice 4 O Choice 5
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I need help with this question. I need to choose from choices 1, 2, 3, 4, 5
![USEFUL FORMULAS
Properties of valid Autocorrelation function (ACF)
The ACF of a signal is defined as the measure of similarity or coherence between a signal and
its time delayed version.
• ACF of a real function x(t): Rxx(T) = Ex(t) x x(t+T)], where T is a variable called time-shift
(delay) parameter
ACF is an even function
ACF reaches its peak at the origin: R(T) = R(0)
ACF of a periodic function is, itself, periodic with the same period
Autocorrelation of the sum of two completely uncorrelated functions (the cross-correlation
is zero for all) is the sum of the autocorrelations of each function separately
ACF is related to the power spectral density via the Fourier transform
Rxx (T-0) = m₂(x); Rxx (T→ Infinity) = [m₁(x)]²
O Choice 1
O Choice 2
Choice 3
Choice 4
O Choice 5](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F699bc471-e6d3-4107-9047-0ba0f591bc36%2F272aa2ea-af09-4735-a258-7b581288763d%2Fk28f1w_processed.jpeg&w=3840&q=75)
Transcribed Image Text:USEFUL FORMULAS
Properties of valid Autocorrelation function (ACF)
The ACF of a signal is defined as the measure of similarity or coherence between a signal and
its time delayed version.
• ACF of a real function x(t): Rxx(T) = Ex(t) x x(t+T)], where T is a variable called time-shift
(delay) parameter
ACF is an even function
ACF reaches its peak at the origin: R(T) = R(0)
ACF of a periodic function is, itself, periodic with the same period
Autocorrelation of the sum of two completely uncorrelated functions (the cross-correlation
is zero for all) is the sum of the autocorrelations of each function separately
ACF is related to the power spectral density via the Fourier transform
Rxx (T-0) = m₂(x); Rxx (T→ Infinity) = [m₁(x)]²
O Choice 1
O Choice 2
Choice 3
Choice 4
O Choice 5
Transcribed Image Text:Check with reasons whether the functions f(t, t+T) listed below represent in each case a valid
autocorrelation function (ACF); if so, why?
Case
(2-ITI)
f(tt+T) (-2sTs+2)
O otherwise
pi= 3.1416
Choices
1
2
3
4
5
(1)
Valid ACF? YES/NO
No:
Not maximum at
T=0
Odd function
No:
(ii)
Reason(s)?
Minimum
at T=0
Yes;
(axt + (1 - (TI)
(-1 sTs+1)
and for all t
O otherwise
a: constant
Yes; Valid
Yes:
Maximum at T = Ely) is
0
Yes; It is an odd
Even symmetry of time
(ii)
function
0
No:
It is a function
of time
independent
No:
Maximum at T = It is a function
O with even
of time
symmetry
No:
No:
Maximum at T = It is a function
of time.
(iii)
(AT+B)/(C|T|+D)
(A, B, C, D):
Constants and
B/D > A/C
(iii)
No:
It is not
an even
function
Yes: Valid
Max at T = 0
Even
symmetry
It is not function Maximum
of time
No:
at T=0
Yes; Even
function
Maximum
at T=0
Yes:
It is not al
function
of time
sin(|T) + 1
(-pi/2 s T s
+pi/2)
0 otherwise
(iv)
Yes;
Yes;
Maximum at T Maximum at T = 0
=0 with
No:
It is a function
(v)
Even symmetry function
of time
(v)
Yes; Valid
Yes; Valid ACF
Even symmetry Not a function of
at the ordinate time. Even function
No: Minimum
at T = 0
It is an even
function
axsin(t) + bxT
(- pi/2 sTs+
pi/2)
0 otherwise
No:
It is a function
of time
It is an even
Yes:
Maximum at T = 0
It has odd
symmetry
No: It is a function
of time
Yes;
It is not a function
of time
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