Linear phase response

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