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The definition of phase of sinusoids signals.
the fraction of the period that has elapsed relative to the origin
The magnitude spectrum of the DTFT shows
The proportion of a complex exponential of a certain frequency should be used to synthesise a signal x[n]
The phase spectrum shows the
initial angle of each complex exponential should be in order to synthesise x[n]
Effect of phase
the information about the shape of the time domain waveform is contained in the phase rather than the magnitude
The effect of an LTI system on a complex exponential
the output to complex exponential is a complex exponential at the frequency of oscillation but a altered magnitude and phase
y[n] = H(Ω)exp(jΩn)
H(Ω) - eigenvalue, exp(jΩn) - eigenfunction
The frequency response of a H(e^(jΩ)) can be expressed as
H(e^(jΩ)) = Re [H(e^(jΩ)) ] + Im[H(e^(jΩ)) ]
which has a magnitude and phase response
Distortionless signal
ensures the shape of the input signal is the same as the output
How to ensure a distortionless response when passing an input to a LTI system
A distortionless signal y[n] = Gx[n-n_d] for G > 0.
In the frequency domain Y(Ω) = Gexp(_jΩn_d)X(Ω)
H(Ω) = Y(Ω) / X(Ω) = Gexp(_jΩn_d)
=>
|H(Ω)| = G
∠H(Ω) = -Ωn_d
The magnitude of the frequency response of the system must be constant i.e flat
The phase of the frequency response must be linear function of frequency
The Real Frequency domain will contain some
Magnitude distortion in the passband due to ripples in the passband
The phase must be constant to ensure a distortionless waveform
No, the phase must be linear not non linear or constant
Phase shift is the
phase response in rads experienced by each sinusoidal component of the input signal
Phase delay is the
is the time shift in the time domain (in number of samples) that is experienced by each sinusoidal component of the input signal.
it is also the phase response divided by the frequency.
In linear phase response, the phase delay ensures no phase distortion
by ensuring all sinusoidal components of the input signal experience the same phase shift
Group Delay
The negative of the slope of the continuous(unwrapped) phase response
arg is the
Argument function
arg(c) =
ARG(c) + 2πr(Ω) where r(Ω) is an integer
The unwrapped phase
continuous phase response evaluated on the unit circle
Ψ(Ω) = arg[H(e^(jΩ))] = ARG(H(e^(jΩ))) + 2πr(Ω)
Wrapped phase is aka
Principal phase
Principal value of the phase response =
ARG(H(e^(jΩ))) for -π< ARG(H(e^(jΩ))) < π
To move from the wrapped phase to the continuous phase
Integer value r(Ω) of 2π i.e 2πr(Ω) must be added to the principal value
Generalised linear phase resp9onse requires a
constant group and phase delay
A zero phase response results in
no group or phase delay , no distortion
A zero phase response implies
a real non-negative frequency response
In zero phase response , what are the effects on the phase of the system
The magnitude of the phase response = real part of the phase response.
Phase of the output = phase of the input (distortionless)
no delay is added to the signal
the spectrum in purely real
is the casual digital filter with zero phase possible
NO because real signals observe symmetry resulting in negative frequencies and a non-casual filter
Casual Linear phase response of FIR filters with group delay can be achieved
filters with conjugate symmetry or anti-symmetry
types of filters with conjugate symmetry or anti-symmetry
symmetric filter coefficients - even symmetry
anti-symmetric filter coefficients - odd symmetry
For even and odd symmetric FIR filters what’s their effect on the coefficients for a filter of order N
Even - > coefficients b_k = b_N-k
Odd - > coefficients b_k = -b_N-k
The symmetric filter has an even symmetry for FIR CASUAL LINEAR PHASE FILTERS in form
H(Ω) = exp(-jNΩ/2) ∑b_kcos((N-2k)Ω/2)
The anti-symmetric filter has an odd symmetry for FIR CASUAL LINEAR PHASE FILTERS
H(Ω) = jexp(-jNΩ/2) ∑b_ksin((N-2k)Ω/2)
How can the restriction on the locations of zeros for symmetry & anti-symmetry on FIR CASUAL LINEAR PHASE FILTERS
By checking the frequency response at Ω = π & Ω = 0 for N being both odd and even
if N is odd it has even length and vise versa
if N is odd, N-2k is odd and if N is even N-2k is even
Symmetric filters restrictions on FIR filters are
If N is odd,
the filter at Ω = 0, the cos(0) = 1 → There are no zeros for any value of the length N+1
the filter at Ω = π, N-2k is odd, the cos(kπ/2) , where k is odd = 0 → There is a zero at Ω = π → z = -1
if N is even,
the filter at Ω = 0, the cos(0) = 1 → There are no zeros for any value of the length N+1
the filter at Ω = π, N-2k is even, the cos(kπ/2), where k is even = -1 → There are no zeros for any value of the length N+1
Anti-symmetric filters restrictions on FIR filters are
If N is odd,
the filter at Ω = 0, the sin(0) = 0 → There is a zero at Ω = 0 → z = 1
the filter at Ω = π, N-2k is odd, the sin(kπ/2) , where k is odd = ± 1 depending on k → no zeros for any value of the length N+1 at Ω = π
if N is even,
the filter at Ω = 0, the sin(0) = 0 → There is a zero at Ω = 0 → z = 1
The filter at Ω = π, N-2k is even, the sin(kπ/2), where k is even = 0 → There is a zero at Ω = π → z = - 1
Symmetric filters with even length can be used as
→ odd order
Low pass filters because of their possession of zeros at z = -1 when N is odd
Antisymmetric filter cannot be used as
Lowpass filters because of both odd and even order they all have zeros at z = 1
Types of linear phase FIR filters based on type of symmetry and length arranged in decreasing flexibility
Type I - Even order symmetric
Type II - Odd order symmetric
Type III - Even order anti symmetric
Type IV -Odd order anti symmetric
Non -casual IIR filtering with zero phase
process input data with a casual real coefficient filter
time reverse the output of the filter
process again by the same filter
time reverse once again the output of the second filter
is the auditory system of humans sensitive to changes in phase
No
Note
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