Radar Systems Analysis and Design Using MATLAB by Mahafza Bassem R.;

Author:Mahafza, Bassem R.;
Language: eng
Format: epub
Publisher: CRC Press LLC
Published: 2022-01-17T00:00:00+00:00

8.8.1 Concept of Stationary Phase

Consider the following bandpass signal

Eq. (8.78)

where ϕ(t) is the frequency modulation. The corresponding analytic signal (pre-envelope) is

Eq. (8.79)

where is the complex envelope and is given by

Eq. (8.80)

The lowpass signal r(t) represents the envelope of the transmitted signal; it is given by

Eq. (8.81)

It follows that the Fourier transform of the signal can then be written as

Eq. (8.82)

Eq. (8.83)

where |X(ω) is the modulus of the Fourier transform and Φ(ω) is the corresponding phase frequency response. It is clear that the integrand is an oscillating function of time varying at a rate of

Eq. (8.84)

The most contribution to the Fourier transform spectrum occurs when this rate of change is minimal. More specifically, it occurs when

Eq. (8.85)

The expression in Eq. (8.85) is parametric since it relates to two independent variables. Thus, for each value ωn there is only one specific ϕʹtn) that satisfies Eq. (8.85). Thus, the time when this phase term is stationary will be different for different values of ωn. Expanding the phase term in Eq. (8.85) about an incremental value tn using Taylor series expansion yields

Eq. (8.86)

An acceptable approximation of Eq. (8.86) is obtained by using the first three terms, provided that the difference (t – tn) is very small. Now, using the right-hand side of Eq. (8.85) into Eq. (8.86), and terminating the expansion at the first three terms yields

Eq. (8.87)

By substituting Eq. (8.87) into Eq. (8.82), and using the fact that r(t) is relatively constant (slow-varying) when compared to the rate at which the carrier signal is varying, gives

Eq. (8.88)

and represent infinitesimal changes about tn. Equation (8.88) can be written as

Eq. (8.89)

Consider the changes of variables

Eq. (8.90)

Eq. (8.91)

Using these changes of variables leads to

Eq. (8.92)

where

Eq. (8.93)

The integral in Eq. (8.92) is of the form of a Fresnel integral, which has an upper limit approximated by

Eq. (8.94)

Substituting Eq. (8.94) into Eq. (8.92) yields

Eq. (8.95)

Thus, for all possible values of ω

Eq. (8.96)

The subscript t was used to indicate the dependency of ω on time.

Using a similar approach that led to Eq. (8.96), an expression for can be obtained. From Eq. (8.83), the signal

Eq. (8.97)

The phase term Ф(ω) is (using Eq. (8.87))

Eq. (8.98)

Differentiating with respect to ω yields

Eq. (8.99)

Using the stationary phase relation in Eq. (8.85) (i.e., ω − ϕ′(t) = 0) yields

Eq. (8.100)

and

Eq. (8.101)

Define the signal group time-delay function as

Eq. (8.102)

then the signal instantaneous frequency is the inverse of Tg(ω). Figure 8.14 shows a drawing illustrating this inverse relationship between the NLFM frequency modulation and the corresponding group time-delay function.

FIGURE 8.14 Matched filter time delay and frequency modulation for a NLFM waveform. Comparison of Eq. (8.97) and Eq. (8.82) indicates that both equations have similar form. Thus, if one substitutes X(ω)/2π for r(t), Ф(ω) for ϕ(t), ω for t, and –t for ω in Eq. (8.82), a similar expression to Eq. (8.95) can be derived. That is,

Eq. (8.103)

The subscript ω was used to indicate the dependency of t on frequency. However, from Eq. (8.80),

Eq. (8.104)

It follows that Eq. (8.103) can be rewritten as

Eq. (8.105)

Substituting Eq. (8.104) into Eq. (8.105) yields a general relationship for any t

Eq.

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