Principles of Modern Radar - Volume 2 1891121537

28 CHAPTER 2 Advanced Pulse Compression Waveform ModulationsLFM waveform include the use of a voltage-controlled oscillator or a digital waveformgenerator. In this example, a heterodyne transmit/receive architecture with two mixerstages is employed. A homodyne (single-stage) receiver or a heterodyne receiver withmore than two stages may also be used.On transmit, the IF signal is mixed to an RF using a second oscillator, which isLO 2,tx = cos (2π f 2 t + θ 2 ) t ≥ 0 (2.2)and is mixed with the signal in equation (1) to produce(x(t) = cos 2π( f 1 + f 2 )t + π β (t − τ ) )2+ θ 1 + θ 2τ 20 ≤ t ≤ τ (2.3)with the lower sideband removed via filtering and the result scaled to unit amplitude. Thesignal is now centered at an RF f 0 = f 1 + f 2 . The transmit signal propagates throughspace and is reflected by targets and clutter.2.2.4.2 ReceiverIn a pulsed system, the receive action is timed to coincide with a specific time delay t rcv ,which is referenced to the center of the range window. Signals reflected toward the radarare mixed on receive with a copy of the transmit modulation. Consider the return from apoint target located at time delay t d(x r (t) = cos 2π( f 1 + f 2 )(t − t d ) + π β ((t − τ ) ) )2− t d + θ 1 + θ 2 t d ≤ t ≤ t d + ττ 2(2.4)The first receive oscillator consists of an intermediate frequency, f 2 , and a linear frequencymodulated component that matches the transmit waveform(LO 1,rcv = cos 2π f 2 t + π β ((t − τ ) ) ) 2− t rcv + θ 2 t rcv ≤ t ≤ t rcv + τ (2.5)τ 2The output of the first mixer is(y(t) = cos 2π f 1 (t − t d ) − 2π f 2 t d + π β ( (−2 t − τ )(t d − t rcv ) + ( td 2 τ2− t rcv2 ) ) )+ θ 1(2.6)where t rcv ≤ t ≤ (t rcv + τ − t d ) for a time delay less than or equal to the center of thereceive window, (i.e., t rcv − τ ≤ t d ≤ t rcv ), and t d ≤ t ≤ (t rcv + τ) for a time delaygreater than or equal to the center of the receive window, (i.e., t rcv ≤ t d ≤ t rcv + τ).The upper sideband associated with the mixer output is removed via filtering, and thesignal is centered at an IF f 1 . Note that the amplitude of the return, which is dependenton the target’s RCS and radar parameters, is suppressed to simplify the expressions. Thestretch processor is linear; thus, the following analysis applies equally to a superpositionof returns from targets at different delays.The signal and timing relationships are illustrated in Figure 2-1. The receive oscillator,in Figure 2-1, is centered at RF for illustrative purposes. The homodyne representation iseasier to depict and yields equivalent baseband results. The receive oscillator is turned onat time t rcv . The returns from two targets are present at time delays t d1 and t d2 , respectively.