VIEWS: 2 PAGES: 22 POSTED ON: 8/22/2011
This deals about MATLAB programs
www.jntuworld.com Features from Time-Frequency Concepts • Time-frequency signal analysis and processing concerns the analysis and processing of signals with time varying frequency content. • Such signals are best represented by a Time-Frequency Distri- bution (TFD). • TFD is intended to show how the energy of the signal is dis- tributed over the two-dimensional time-frequency space. • Processing of signal may then exploit the features produced by the concentration of signal energy in two dimensions(time and frequency) instead of only one (time or frequency). www.jntuworld.com Mono-component and Multi-component Signals: •A mono-component signal is described in the (t, f ) domain by one single ridge corresponding to an elongated region of energy con- centration. • Furthermore, interpreting the crest of the ridge as a graph of instantaneous frequency (IF) versus time, we require the IF of a mono-component signal to be a single valued function of time. •A multi-component signal may be described as the sum of two or more mono-component signals. www.jntuworld.com Instantaneous Frequency (IF) and Mean Frequency: • The Instantaneous frequency of a mono-component signal is 1 ′ fi (t) = φ (t) 2π where φ(t) is the instantaneous phase of the signal s(t). •A multi-component signal will have a separate IF for each com- ponent. • The IF is a detailed description of the frequency characteristics of a signal. • This contrasts with the deﬁnition of mean frequency deﬁned next. • The mean frequency of a signal is ∞ 0 f. | S(f ) |2 df fo = ∞ 0 | S(f ) |2 df where S(f ) is the Fourier Transform of the signal s(t). www.jntuworld.com • f0 is the ﬁrst moment of | S(f ) |2 with respect to frequency: i.e., we deﬁne the mean frequency as if | S(f ) |2 were the probability density function of the frequency. Instantaneous Frequency and Time Delay: • Instantaneous frequency of a signal indicates the dominant fre- quency of the signal at a given time. • Let us now seek the dual of the IF, indicating the dominant time when a given frequency occurs. • If z(t) is an analytic signal with the Fourier Transform Z(f ) = Aδ(f − fi) where A is in general complex, the dominant frequency is fi. www.jntuworld.com • Taking the inverse Fourier transform of Z(f ) gives z(t) = Aej2πfi t • The instantaneous phase of z(t), denoted by φ(t), is φ(t) = arg(z(t)) = 2πfit + arg(A) so that fi = 2π φ′(t) 1 • Now let us repeat the argument with the time and frequency variables interchanged. • The signal z(t) = aδ(t − τd) where a is in general complex, is an impulse at τd. • The Fourier transform of the signal is Z(f ) = ae−j2πf τd www.jntuworld.com and its phase, denoted by θ(f ), is θ(f ) = arg(Z(f )) = −2πf τd + arg(a) so that τd = − 2π θ′(f ). τd is termed as time delay. 1 • If z(t) is an analytic signal with the Fourier transform Z(f ), then the time delay (TD) of Z(f ), denoted by τd(f ), is 1 ′ τd(f ) = − θ (f ) 2π where θ(f ) = arg(Z(f )). • Notice that the deﬁnitions of τd and fi are similar except that time and frequency are interchanged and the time delay has an extra minus sign, hence time delay is termed as the dual of instanta- neous frequency. www.jntuworld.com Mean Instantaneous Frequency and Group Delay: • Let z(t) be a band pass analytic signal with center frequency fc. • Let its Fourier transform be Z(f ) = M (f − fc )ejθ(f ) where the magnitude M (f − fc), and phase θ(f ) are real. • If the signal has linear phase in the support of Z(f ), i.e., if θ(f ) is a linear function of f whenever Z(f ) is non-zero, we can let θ(f ) = −2πτpfc − 2πτg [f − fc] where τ p and τg are real coeﬃcients with dimensions of time. Z(f ) = M (f − fc )e−j(2πτpfc+2πτg [f −fc]) = e−j2πτpfc .M (f − fc ).e−j2πτg [f −fc] www.jntuworld.com • Taking the Inverse Fourier transform of Z(f ), we ﬁnd z(t) = m(t − τg ).e2πfc(t−τp ) where m(t) is the Inverse Fourier transform of M (f ). • Now because M (f ) is real, m(t) is Hermitian, [i.e., m(−t) = m∗(t)], so that | m(t) | is even. • Hence τg is the time about which the envelope function is sym- metrical, for this reason,τg is called the group delay. • The phase of the oscillatory factor is −2πfcτp, where τp is called the phase delay. • If an analytic signal has the Fourier transform Z(f ) =| Z(f ) | ejθ(f ) www.jntuworld.com then the Group Delay(GD) of the signal is 1 ′ τg (f ) = − θ (f ) 2π and the phase delay of the signal is 1 τp(f ) = − θ(f ) 2πf • The relation for time delay (τd) and group delay are same. However the physical interpretations are diﬀerent. • The time delay applies to an impulse, where as the group delay applies to the envelope of a narrow band signal. • Now consider the dual of the above. • Let z(t) be a time limited signal centered on t = tc, and let z(t) = a(t − tc)ejφ(t) www.jntuworld.com where a(t) and φ(t) are real. If the signal has constant in- stantaneous frequency in the support of z(t), i.e., if φ(t) is a linear function of t wherever z(t) is non zero, we can let, φ(t) = 2πf0tc + 2πfm [t − tc] where f0 and fm are real coeﬃcients with dimensions of fre- quency. z(t) = a(t − tc)ej(2πf0tc +2πfm[t−tc ]) = ej2πf0tc .a(t − tc).ej2πfm[t−tc ] • Taking the Fourier transform of z(t) we ﬁnd Z(f ) = A(f − fm ).e−j2π[f −f0 ]tc where A(f ) is the Fourier transform of a(t). • Now a(t) is real, A(f ) is Hermitian, so that | A(f ) | is even. www.jntuworld.com • Hence fm is the frequency about which the amplitude spectrum is symmetrical; for this reason, fm is called mean instantaneous frequency. • For the signal, z(t) =| z(t) | ejφ(t), the mean instantaneous frequency is 1 ′ fm (t) = φ (t) 2π • Thus the mean instantaneous frequency is same as the instanta- neous frequency deﬁned earlier, but the physical interpretations are diﬀerent. • The instantaneous frequency has been derived for a tone, where as the mean instantaneous frequency applies to the spectrum of a short duration signal. www.jntuworld.com Heuristic Formulation of Time-Frequency Distributions • Given an analytic signal z(t) obtained from a real signal s(t), we seek to construct a time-frequency distribution ρz (t, f ) to represent precisely the energy, temporal and spectral characteristics of the signal. The Time-Frequency Distribution: • For a monocomponent signal, it is reasonable to expect that the TFD should take the form of a knife-edge ridge whose crest is a graph of the IF law in the (t, f ) plane. www.jntuworld.com • Mathematically, the knife-edge is idealized as a delta function with respect to frequency. Knifes-Edge IF Indication: • Noting that ρz is a function of frequency f and represents a kind of spectrum, we may reasonably require ρz to be Fourier transform of some function related to the signal. • Let us call this function as Signal Kernel and give it the symbol Kz . • So the signal kernel can be written as Kz (t, τ ), and the TFD is ρz (t, f ) = F {Kz (t, τ )} τ →f • To ﬁnd a suitable form for Kz (t, τ ), for simplicity, let us ﬁrst consider the case of unit amplitude mono component FM signal. z(t) = ejφ(t) www.jntuworld.com • Whose instantaneous frequency is φ′ (t) fi (t) = 2π • We would like the TFD of z(t) at any given time to be a unit delta function at the instantaneous frequency, so the ”instanta- neous frequency” reduces to the ordinary Fourier transform in the constant frequency case. • That is, we want ρz (t, f ) = δ(f − fi (t)) • Substituting into the deﬁnition ρz (t, f ) we have δ(f − fi (t)) = F {Kz (t, τ )} τ →f ′ Kz (t, τ ) = F −1{δ(f − fi (t))} = ej2πfi(t)τ = ejφ (t)τ τ ←f www.jntuworld.com • Because φ′(t) is not directly available, we write φ(t + τ /2) − φ(t − τ /2) φ′ (t) = lim τ →0 τ and use the approximation 1 φ′ (t) ≈ [φ(t + τ /2) − φ(t − τ /2)] τ which is called the central ﬁnite diﬀerence (CFD) approximation. Substituting φ′(t) into Kz (t, τ ) gives the signal kernel. Kz (t, τ ) = ejφ(t+τ /2).e−jφ((t−τ /2)) Kz (t, τ ) = z(t + τ /2).z ∗ (t − τ /2) The Wigner Distribution: • Substituting Kz (t, τ ) in ρz (t, f ) we get ρz (t, f ) = F {z(t + τ /2).z ∗(t − τ /2)} τ →f www.jntuworld.com ∞ ρz (t, f ) = z(t + τ /2).z ∗(t − τ /2)e−j2πf τ dτ −∞ • The above equation is given the symbol Wz (t, f ) and is called the Wigner Distribution (WD). 1 • The approximation φ′(t) ≈ τ [φ(t + τ /2) − φ(t − τ /2)] is exact if φ is a linear function of t, i.e. if φ′(t) is constant. • It is also exact if φ(t) is quadrature, i.e. if φ(t) is linear. • Thus the WD gives an unbiased estimate of the IF for a complex linear FM signal. • The constant frequency real signal s(t) = cos(2πfct) leads to the kernel Ks(t, τ ) = s(t + τ /2)s(t − τ /2) www.jntuworld.com = cos(2πfc(t + τ /2)cos(2πfc(t − τ /2) 1 1 = cos(2πfc(2t)) + cos(2πfcτ ) 2 2 Note:cos(A + B) + cos(A − B) = 2cos(A)cos(B) Let A = t + τ /2 & B = t − τ /2 cos(2t) + cos(τ ) • Taking Fourier transforms with respect to τ gives the WD 1 1 1 Ws(t, f ) = δ(f − fc) + δ(f + fc) + cos(2πfc(2t))δ(f ) 4 4 2 • The terms in δ(f ∓ fc) are naturally expected and arise because s(t) may be expressed as a sum of complex sinusoids at frequencies ±fc. • The term in δ(f ) is an artifact arising because the non linearity of the WD causes interaction between the positive and negative frequency terms. www.jntuworld.com • In a similar way, we can ﬁnd the Ks(t, τ ) for a non-windowed linear FM signal α s(t) = Acos(2π(f0t + 2t2 )) Ks(t, τ ) = s(t + τ /2)s(t − τ /2) s(t + τ /2) = Acos(2π(f0(t + τ /2) + α (t + τ /2)2)) 2 s(t − τ /2) = Acos(2π(f0(t − τ /2) + α (t − τ /2)2)) 2 A = f0(t + τ /2) + α (t + τ /2)2) 2 B = f0(t − τ /2) + α (t − τ /2)2) 2 2cos(A)cos(B) = cos(A + B) + cos(A − B) cos(A + B) = cos(2π(f0(t + τ /2) + α (t + τ /2)2 + f0(t − τ /2) + α (t − τ /2)2)) 2 2 2 2 = cos(2π((f0t + f0 τ + α t2 + α τ4 + 2. α .t. τ ) + (f0t − f0 τ + α t2 + α τ4 − 2. α .t. τ ))) 2 2 2 2 2 2 2 2 2 2 www.jntuworld.com 2 = cos(2π(2f0t + αt2 + α τ4 )) cos(A − B) = cos(2π(f0(t + τ /2) + α (t + τ /2)2 − f0(t − τ /2) + α (t − τ /2)2)) 2 2 2 2 = cos(2π((f0t + f0 τ + α t2 + α τ4 + 2. α .t. τ ) − (f0t − f0 τ + α t2 + α τ4 − 2. α .t. τ ))) 2 2 2 2 2 2 2 2 2 2 = cos(2π(f0τ + αtτ )) = cos(2π((f0 + αt)τ )) = cos(2π(fi(t)τ )) when fi(t) = f0 + αt 2 Ks(t, τ ) = 1 A2 cos(2π(fi(t)τ )) + 1 A2 cos(2π(2f0t + αt2 + α τ4 )) 2 2 • Taking Fourier transform with respect to τ gives the WD Ws(t, f ) = F {Ks(t, τ )} τ →f www.jntuworld.com 2 = 1 A2δ(f − fi (t)) + 2 A2δ(f + fi (t)) + 1 A2 F {cos(2π(2f0t + αt2 + α τ4 ))} 4 1 2 τ →f • The terms in δ(f ∓ fi(t)) are naturally expected, while the last term in the signal kernel gives rise to a continuum of artifacts in the WD. • These artifacts, which greatly diminish the usefulness of the WD for real signals, are removed by modifying the WD with the ana- lytic signal, which is termed as Wigner-Ville Distribution. The Wigner-Ville Distribution (WVD): • The Wigner-ville Distribution (WVD) of a signal s(t), denoted by Wz (t, f ), is deﬁned as the WD of its analytic associate, i.e., Wz (t, f ) = F {z(t + τ /2).z ∗(t − τ /2)} τ →f where z(t) is the analytic associate of s(t). www.jntuworld.com • The name ‘Wigner-Ville distribution”, as opposed to ‘Wigner dis- tribution”, emphasizes the use of the analytic signal. • Wigner proposed WD in the context of quantum mechanics. • Ville ported WD in a signal processing context. • Noting that a signal can have a time-dependent frequency con- tent, Ville sought an instantaneous spectrum having the attributes of an energy density and satisfying the so-called marginal condi- tions. • The integral of the TFD Wz (t, f ) with respect to frequency is the instantaneous power |z(t)|2. • The integral of the TFD Wz (t, f ) with respect to time is the energy spectrum |Z(f )|2. • We know that, www.jntuworld.com Wz (t, f ) = F {Kz (t, τ )} τ →f • The signal kernel Kz (t, τ ) is also called the instantaneous auto cor- relation function (IAF) of z(t).