Receiver System .1 Matched Filter.1Matched Filter

Noise power increases and the SNR deteriorates if the receiver bandwidth B is wider than that of the bandwidth of the received signals. On the other hand, if B is narrower than the required bandwidth to receive an relatively undistorted echo from a point scatterer, noise power decreases and range resolution worsens. However, the signal power decreases too, causing distortion and lack of information of signals. If it is constrained that the resolution to be a constant, the SNR is an optimum when the filter response is matched to the transmitted pulse (Doviak and Zrni´c 2006, p. 83).

Therefore, it is crucial to determine B by keeping the balance between the bandwidth which the desired received signals occupy and that which makes SNR best. The optimum solution for this subject is the filter bandwidth which maximizes SNR to the received signals for pulse radar, and the filter is called matched filter.

Response of Matched Filter

The filter characteristic (transfer function) which maximizes SNR are discussed in this paragraph (Nathanson 1991, pp. 355–360). If the time dependence of the signal at the receiver input is s(t), the total energy E which is included in s(t) is expressed as

E=

 _∞

−∞s²(t)dt =

 _∞

−∞|S( f )|²d f, (5.37)

where E should be finite. The complex amplitude spectrum S( f ) is the Fourier transform⁸of s(t) and given by

S( f ) =^{ ∞}

−∞s(t)e^{− j2πf t}dt. (5.38) The filter output signal so(t) is expressed by S( f ) and the filter transfer function H( f ) as

so(t) =^{ ∞}

−∞S( f )H( f )e^{j2πf t}d f. (5.39) Meanwhile, the power spectral density (or simply power spectrum)⁹of output signal for linear filter becomes a product between the power spectrum of filter input signal and the squared absolute value of filter transfer function. Thus, using noise power spectral density N0/2 from (5.1), the noise power of the filter output PNis given by

PN=N0

2

 _∞

−∞|H( f )|²d f. (5.40)

The optimum condition for receiver filter, i.e., the transfer function which maximizes SNR is one which maximizes the ratio of the maximum value of the signal to noise power at the output port. From (5.39) and (5.40), the value of SNR is given bySkolnik(2001, p. 279)

|so(tM)|²

where t_Mdenotes the time t at which the output|so(t)|²is a maximum. The function

|H( f )| which maximizes (5.41) is that of the matched filter, and is derived by applying Schwarz’s inequality to two complex functions a(x) and b(x) as follows.

^_−∞^∞a(x)b(x)dx

8The Fourier transform will be described in Sect.5.4.1.

9Power spectrum will be defined in Sect.5.4.2.

is obtained. The equality condition applies only when the ratio of a(x) to b^∗(x) is constant, or H( f ) and S( f ) are in the following relation (Skolnik 2001, p. 280).

H( f ) = KS^∗( f ) e^{− j2πf tM}, (5.44) where K is an arbitrary gain constant, and the superscript ^∗ denotes complex conjugate. The filter whose transfer function has the characteristics of (5.44) is the matched filter, and (5.43) becomes

|so(tM)|² PN =2E

N0, (5.45)

where |so(tM)|² is the peak power of signal, and PN is noise power. As shown in (5.45), the ratio of the peak signal power to noise power for the matched filter output depends only on the spectral densities of signal power which is the infinite integral of power spectra of receiver input and noise. It should be noted that the ratio does not depend on the signal waveform. The time domain expression for H( f ) is the inverse Fourier transform¹⁰of (5.44), and expressed as

h(t) = K

 _∞

−∞S^∗( f ) e^{− j2πf(tM−t)}d f= Ks^∗(tM−t). (5.46) Equation (5.46) indicates that the impulse response of a matched filter is the same as that for delayed input signal on the reversed time axis. Here, it should be noted that a matched filter maximizes SNR of output signal; however, it does not reproduce the waveform of input signal.

Matched Filter and Correlation Function

The correlation function and the impulse response of a linear system are utilized to explain (5.46). Here, we simplify K = 1. The cross-correlation function of two different signals x1(t) and x2(t) for finite delay time τ is defined as follows for signals whose energy is finite as shown by

Rx₁x₂(τ) =^{ ∞}

−∞x1(t)x2∗(t −τ)dt. (5.47) In signal analysis, the integration in the form of (5.47) is often called correlation function. However, it is applicable only to the signal that the total energy of (5.37) is finite. In general, correlation function indicates the one to be described in Sect.5.4.2, and thus it is necessary not to confuse both.

10The inverse Fourier transform will be discussed later in Sect.5.4.1as the Fourier transform.

When the input signal yi(t) is applied to the filter whose impulse response is h(t), the output signal yo(t) is obtained as the response of the linear system

yo(t) =

 _∞

−∞yi(τ)h(t −τ)dτ. (5.48) If the filter is matched filter, h(t) = s^∗(tM−t) from (5.46), and consequently (5.48) is expressed as

yo(t) =^{ ∞}

−∞yi(τ)s^∗[τ− (t −tM)]dτ. (5.49) Using the relation of (5.47), (5.49) is written as

yo(t) = Rys(t −tM). (5.50) The above equation means that the matched filter output y0(t) is given by the cross-correlation function of input signal yi(t) and the receiver input pulse waveform s(t). In the case that the receiver noise can be ignored, y0(t) is equivalent to autocorrelation function of receiver input pulse waveform.

5.2.2 Frequency Conversion and Phase Measurement

Generally, the received signal is detected as continuous signal. There are two meth-ods for detection; one is the envelope detection that treats only amplitude of signals, and the other is synchronous detection which treats both amplitude and phase of signals. In synchronous detection, it is necessary to keep the phase information of the transmitted signal until the scattered signal is received. Consequently, the system configuration for synchronous detection becomes more complicated than that for amplitude detection. The synchronous detection, however, enables to obtain the power spectrum which contains the wind velocity information in addition to the echo intensity information. Therefore, recent meteorological and atmospheric radars almost universally adopt synchronous detection. The choice of the carrier frequency for practical operational meteorological and atmospheric radars depends principally on the type of scatterers to be detected (e.g., Bragg, insects, cloud particles, precipitation particles, etc.), the needed angular resolution, and cost. The carrier frequencies range from several MHz to 100 GHz (Table1.1). Because signal processing cannot, for most meteorological and atmospheric radars, be performed on signals at the carrier frequency (especially if the carrier frequency f0is above 1 GHz), the received signal (at the carrier frequency f0) is down converted to a lower IF from several MHz to approximately 100 MHz. The phase information of the received signal whose carrier frequency is down converted to an IF frequency is preserved, and thus the phase of the transmitted pulse can be compared to the phase

Fig. 5.6 The process for frequency conversion of radar received signals and phase measurements.

The transmitted carrier frequency is denoted by f0, while the local (LO) and the IF by fsand fc, respectively

of the received IF signal. The typical scheme to compare the phase of the transmitted pulse and that of the received signal is to generate both the carrier frequency of transmitted pulse and receiver frequency necessary for phase measurement¹¹from a common standard signal, i.e., the coherent oscillator (COHO) in Fig.5.6. In general, the desired carrier frequency of transmitted pulse is generated by up-converting the COHO frequency to f0, whereas, the COHO signal is used as is for phase measurement.

A basic process for the frequency conversion of received signals and phase measurements is illustrated in Fig.5.6. The local frequency (LO) signal with frequency fs is generated by a highly stabilized local oscillator (STALO). The LO signal is mixed with the IF signal with frequency fc which is generated by the COHO, and becomes f0= fs+ fc where f0is the transmitted frequency. The up converted continuous wave (CW) at f0 is pulse modulated and amplified by amplifying devices such as transistor and klystron, and sent to antenna. Here the CW is not the “carrier” before pulse modulation because it is not carrying any information. It becomes “carrier” only after pulse modulation.

The frequency of the received signal from a moving scatterer is shifted by an amount fd due to Doppler effect and becomes f0+ fd. When this signal is mixed with LO frequency fs, an IF signal with fc+ fdthat possesses the phase information of the scatterer is generated. Subsequently, the IF signal is mixed with the COHO frequency to detect phase, and Doppler frequency fdis obtained by measuring the change of the echo’s phase angle from one transmitted pulse to the next as was discussed in Sect.4.1.1. From (4.11), it is derived that the maximum measurable Doppler frequency fdmaxis the half of the pulse repetition frequency (PRF), thus the values of fdis between 0 and around several hundred Hz to kHz depending on the PRF of the radar.

11The phase measurement function in synchronous detection.

A received signal x(t) at time t is expressed by operational (or carrier) frequency f0, amplitude a(t), and phaseϕ(t) as

x(t) = a(t)cos[2πf0t+ϕ(t)]. (5.51) Phaseωdt= 2πfdt in (4.9) is generalized toϕ(t). The complex signal s(t), which corresponds to real signal x(t) in (5.51), is given by

s(t) = a(t)e^{j[2πf0t+ϕ(t)]}. (5.52) As described in Sect.4.1.1, the complex received signal is expressed as follows by performing frequency conversion for s(t) and removing carrier frequency term,

v(t) = a(t)e^jϕ(t), (5.53)

where v(t) is the complex envelope signal of s(t). If in-phase and quadrature-phase signals are expressed as

I(t) = a(t)cosϕ(t) , Q(t) = a(t)sinϕ(t), (5.54) respectively, (5.53) becomes

v(t) = I(t) + jQ(t). (5.55)

As the basic function of a Doppler radar which is shown in Fig.4.1, the procedure to convert received signal into IF signal and subsequently to obtain I and Q signals by synchronous detection is shown in Fig.5.6. Two kinds of signals, I and Q, with the phase difference 90^◦, are given as the output of bipolar receiver. The I(r,t) and Q(r,t) terms previously shown in (4.6) and (4.7), respectively, are equivalent to the I and Q signals which mean the real and imaginary part of the received envelope signal, respectively.