2. Time Frequency Representations
2.7. Spectrogram
The spectrogram, which is so often used in signal analysis, is in fact just a glorified special STFT of which we only take the square of the TFR to obtain the the energy density of spectrum or the signal probability distribution. The energy density is always the square of the spectrum. This is directly related to the square of the electric field being equal to the energy or intensity of a electromagnetic field or signal. Here we quickly reproduce the STFT definition with a different notation to emphasize that it is just a Fourier transform of a moving average of a windowed function. The window function, w(t), is just multiplied to the signal to generate,
sw(t)(τ ) = s(τ )w(τ − t). (2.39)
The window function has limited support, i.e., it is defined only on a finite time, e.g., t ∈ [−T2,T2], so that generally for a hat function (cf. Eq. (A.76) and Eq. (A.80)),
wT(t) = πT(t) = ( 1 if t ∈ [−T 2, T 2] 0 otherwise . (2.40)
The Fourier transform will reproduce the frequency content of the small window of the signal,
Fnsw(t)(τ )
o
= Sw(t)(ν) = Fs(t, ν; w). (2.41)
The STFT contains both amplitude and phase information in the time-frequency domain.
The energy density spectrum at time t, also known as the spectrogram, is simply given by,
P (t, ν) = |St(ν)|
2
. (2.42)
This operation essentially destroys the phase information contained in the STFT. The energy density of the window function is given by,
Ew =
ˆ ∞ −∞
2.7 Spectrogram
The characteristic function is straightforwardly obtained from the definition,
M (θ, τ ) = ˆ ˆ |St(ν)|2eiθt+iτ νdνdt, (2.43) = As(θ, τ )Aw(−θ, τ ), (2.44) where, As(θ, τ ) = ˆ s∗(t − 1 2τ )s(t + 1 2τ )e iθt dt, (2.45) Aw(θ, τ ) = ˆ w∗(t − 1 2τ )w(t + 1 2τ )e iθt dt, (2.46)
are the ambiguity functions (see Subsection 2.8.4) of the signal and the window.
2.7.1. Time-frequency Resolution
Having considered the analogy with filtering theory we can extend these thoughts further by using the Dirac-δ (See Subsection A.4.6) time signal and frequency in- puts to obtain the characteristics of the filter, i.e.,
• Dirac-δ time signal.
s(t) = δ(t − to)
F
⇐⇒ Fs(t, ν; w) = w(t − to) e−2πiνto (2.47)
• Dirac-δ frequency signal (Modulation).
s(t) = e2πiνot ⇐⇒ FF
s(t, ν; w) = W (ν − νo) e−2πiνot (2.48)
Here we clearly notice that the temporal resolution of the STFT is dependent on the duration, h(t), of the analysis window, and the frequency resolution is proportional on the effective bandwidth, H, of the analysis window, h. Thus we are confronted with a reciprocal trade-off between temporal and frequency resolution which is directly inherited from the corresponding Fourier relationship and the Heisenberg-Gabor inequality.
Chapter 2 Time Frequency Representations
2.7.2. Discrete Time-Frequency Representations
Alas all of the above is defined for continuous signals. In the calculation of these transforms the signal had to be discretized. This is the field of Digital Signal Pro- cessing (DSP) [80, 81]. For an example of a discrete Gabor TFR consult [74]. As some background reference, the important sampling theorem is given in Section A.7 along with some properties of band-limited signals in Section A.8. Related to the sampling theorem is the Poisson summation theorem and the Nyquist-Shannon sampling theorem given in Section A.9 and Section A.10. The sampling of a signal is accomplished by a so-called Dirac comb defined in Subsection A.4.6.5. A dis- crete sampled signal3, s[n], with sampling period, ∆t, the sampling period of the
discrete STFT must chosen such that, T = k∆t, with k ∈ N. The discrete STFT for a discrete rectangular window, m, n ∈ Z, with sampling steps sizes, T , and, Ω, becomes,
Fs[n, m; h] = Fs(nT, mΩ; h) =
ˆ ∞
−∞
h∗(τ − nT )s(τ )e−2πimΩτdτ (2.49)
For discrete time-frequency representations Gabor suggested that an arbitrary time signal can be decomposed as,
s(t) =X
n,m
cn,mhn,m(t) hn,m(t) = h(t − mT )einΩt, (2.50)
where the time-frequency domain, (t, ω)∈ R2, is discretized in a lattice were, T and Ω, are the time and frequency lattice intervals. Gabor proposed that the function,
h(t), should coincide with the minimum variance function that is most compact for
the time-frequency bandwidth product which is of course the Gaussian function,
h(t) = (α/π)1/4e−αt2/2. (2.51) The Gabor atoms, hn,m(t) = h(t−mT )einΩt, with both, T and Ω, small, collectively
form an oversampled overcomplete non-orthgonal basis, which is called a frame. It has been shown that such an expansion is possible, if,
Ω × T ≥ 1. (2.52)
Without loosing any signal information, the problem reduces to choosing the values of, T and Ω, so as to minimize the redundancy (overlapping of atoms). The Balian- Low theorem, however, proves that it is impossible to have a window function, h,
2.7 Spectrogram
that is well localized in both frequency and time, and is appropriately named the Balian-Low obstruction. A well localized window, h, (for example a Gaussian window), the reconstruction (synthesis) formula will be numerically unstable. In the discrete case the reconstruction (synthesis) of the signal can generally be written as,
s(t) =X
n,m
Fs[n, m; h]gn,m(t) gn,m(t) = g(t − mT )einΩt, (2.53)
with, h, g, and, T, Ω, subject to 1 Ω X n g(t + k Ω − nT )h ∗(t − nT ) = δ k ∀t, (2.54)
where, δk = 0 ∀k 6= 0 and δo = 1. This discrete condition on the analysis and
synthesis window functions is a much more severe condition than the compliance of the continuous condition.
2.7.3. The Short-Frequency Time Transform.
In motivating the short-time Fourier transform we emphasized the desire to study frequency properties at time t. Conversely, we may wish to study time properties at a particular frequency. We just multiply the spectrum, S(ν), with a frequency window, W (ν), and take the time transform, which of course, is the inverse Fourier transform. In particular, we define the short-frequency time transform by,
sν(t) =
ˆ ∞
−∞
e2πiθtS(θ)W (ν − θ)dθ (2.55)
where of course, w ⇐⇒ WF and s⇐⇒ S, then,F
St(ν) = e−2πiνtsν(t). (2.56)
Only the modulation phase factor, e−2πiνt, separates the the short-frequency time transform from short-time Fourier transform. Since the distribution is the absolute square, the phase factor does not enter into it and either the short-time Fourier transform or the short-frequency time transform can be used to define the joint distribution, P (t, ν) = |St(ν)| 2 = |sν(t)| 2 . (2.57)
This exhibits the desirable fact that the spectrogram can be used to study the behavior of time properties at a particular frequency. This is done by choosing an,
H(ν), that is narrow, or equivalently, due to reciprocity, by taking an, h(t), that
Chapter 2 Time Frequency Representations
2.7.3.1. Narrowband and Wideband Spectrogram.
Due to the reciprocity theorem of Fourier transform (cf. Subsection A.3.4), which manifests itself in the Heisenberg-Gabor uncertainty, if the time window is of short duration, the frequency window, H(ν), is broad; in that case the spectrogram is called a broadband spectrogram. Conversely, if the window is of long duration, then, H(ν), is narrow, and we say we have a narrowband spectrogram.
2.7.4. Characteristic Function.
The characteristic function of the spectrogram is straightforwardly obtained,
MSP(θ, τ ) = ˆ ˆ |St(ω)|2ejθt+jτ ωdtdω (2.58) = As(θ, τ )Ah(−θ, τ ), (2.59) where, As(θ, τ ) = ˆ s∗(t − 1 2τ )s(t + 1 2τ )e jθtdt, (2.60)
is the ambiguity function (see Subsection 2.8.4) of the signal [69], and, Ah, is
the ambiguity function of the window defined in the identical manner, except that we use, h(t), instead of s(t). Note that the ambiguity function has the following property, A(−θ, τ ) = A∗(θ, −τ ).