# Time Frequency TIME DOMAIN PLOT FREQUENCY DOMAIN Carrier at GHz RF Carrier e PDF document - DocSlides

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g 10 GHz Time Frequency TIME DOMAIN PLOT FREQUENCY DOMAIN RF Carrier eg 10 GHz eg 5 GHz 10 GHz Occurs from to t Occurs from to t Time Frequency TIME DOMAIN PLOT FREQUENCY DOMAIN RF Carrier F eg 10 GHz GHz Amplitude Modulation Envelope Detected Signa ID: 22181

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Time Frequency TIME DOMAIN PLOT FREQUENCY DOMAIN Carrier at 10 GHz RF Carrier (e.g. 10 GHz) Time Frequency TIME DOMAIN PLOT FREQUENCY DOMAIN RF Carrier e.g. 10 GHz e.g. 5 GHz 10 GHz Occurs from to t Occurs from to t Time Frequency TIME DOMAIN PLOT FREQUENCY DOMAIN RF Carrier (F ), e.g. 10 GHz GHz Amplitude Modulation Envelope Detected Signal (F AM ), e.g. 100 Hz Upper Sideband Lower Sideband 9,999,999,900 Hz 10,000,000,100 Hz 10 GHz Time Frequency TIME DOMAIN PLOT FREQUENCY DOMAIN Carrier at 10 GHz RF Carrier Detected Signal Square Wave AM Envelope Lower Sidebands Upper Sidebands Carrier Amplitude Modulated by a Square Wave 2-11.1 Figure 1 . Unmodulated RF Signal Figure 2 . RF Signal with Frequency Modulation Figure 3 . Sinewave Modulated RF Signal Figure 4 . Square Wave Modulated RF Signal (50% Duty Cycle AM) MODULATION Modulation is the process whereby some characteristic of one wave is v aried in accordance with some characteristic of another wave. The basic types of modulation are angular mo dulation (including the special cases of phase and frequency modulation) and amplitude modulation. I missile radars, it is common practice t ampl itude modulate the transmitted RF carrie wave of tracking and guidance transmitters b using a pulsed wave for modulating, and t frequency modulate the transmitted RF carrie wave of illuminator transmitters by using a sine wave. Frequency Modulation (FM) - As s hown in Figure 1, an unmodulated RF signal in th time domain has only a single spectral line at the carrier frequency (f in the frequency domain. If the signal is frequency modulated, as shown i Fi gure 2, the spectral line will correspondingl shift in the frequency domain. Amp litude Modulation (AM) - I the signal in Figure 1 is amplitud modul ated by a sinewave as shown i Fi gure 3, sidebands are produced in th frequency domain at F ± F . AM othe AM than by a pure sine wave will caus addi tional sidebands normally at F nF , where n equals 1, 2, 3, 4, etc. AM Pulse mod ulation is a special case of AM wherein the carrier frequency is gated at a pulsed rate. When th reciprocal of the duty cycle of the AM i s a whole number, harmonics corresponding to multiples of that whole number will be missing, e.g. in a 33.33% duty cycle, AM wave will miss the 3rd, 6th, 9th, etc harmonics, while a square wave or 50 dut cycle triangular wave will miss th 2nd, 4th, 6th, etc. harmonic, as shown i Figu re 4. It has sidebands in the frequency domain at F ± nF , wh ere n = 1, 3, 5, etc. AM e amplitude of the power level follows a sine x / x type distribution.

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RF Pulse Modulating Pulse Time Pulse Width T PRI 1/PRF 2/PW 1/PW Frequency Spectrum Envelope 1/PRI 1/PW 2/PW Frequency 3/PW -1/PW -2/PW -3/PW Spectral Line Spacing 1/PRI Amplitude changes from + to - at every 1/PW interval Note: 2nd, 4th, 6th, etc, harmonics are missing , i.e. zero amplitude 1/PRI 1/PW 2/PW Frequency 3/PW -1/PW -2/PW -3/PW Spectral Line Spacing 1/ PRI Amplitude changes from + to - at every 1/ PW interval Note: 3rd, 6th, 9th, etc., harmonics are missing, i.e. zero amplitude Fundamental 3rd Harmonic 5th Harmonic Resultant sin where pulse width PW period PRI and Amplitude of rectangular pulse 2-11.2 Figure 5 . Pulse Width and PRI/PRF Waveforms Figure 6 . Sidelobes Generated by Pulse Modulation (Absolute Value) Figure 7 . Spectral Lines for a Square Wave Modulated Signal Figure 8 . Spectral Lines for a 33.3% Duty Cycle Figure 9 . Square Wave Consisting of Sinewave Harmonics Fi gure 5 shows the pulse width (PW) in the time domain which defines the lobe width in the frequency domai (Figure 6). The w idth of the main lobe is 2/PW, whereas the width of a side lobe is 1/PW. Figure 5 also shows the pulse repetition inter val (PRI) or its reciprocal, pulse repetition frequency (PRF), in the time domain. In the frequency domain, the spectral lines inside the lobes are sep arated by the PRF or 1/PRI, as shown in Figures 7 and 8. Note that Figures 7 and 8 show actual magnitude of the side lobes, whereas in Figure 4 and 6, the absolute value is shown. The magnitude of each spectral component for a rectangular pulse can be determined from the following formula: [1] Figure 7 sh ows the spectral lines for a square wave (50% duty cycle), while Figure 8 shows the spectral lines for a 33.33% duty cycle rectangular wave signal. Figure 9 shows that for square wave A M, a significant portion of the component modulation is contained in the first few harmonics which comprise the wave. There are twice as many sidebands or spectral lines as there are harmonics (one on the plus and one on the minus side of the carrier). Eac sideb and represents a sine wave at a frequency equal to th difference between the spectral line and f .

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Frequency FREQUENCY DOMAIN Frequency Frequency 10 GHz 14 kHz Reflection of a stationary 10 GHz radar from a stationary target such as a metallic balloon. Reflection from a target such as a glider moving at 400 kts toward a stationary 10 GHz radar. Reflection from a jet or prop target moving at 400 kts toward a stationary 10 GHz radar. 10 GHz 2-11.3 Figure 10 . Doppler Return and JEM figure similar to Figure 9 can be created for any rectangular wave. The relative amplitude of the time domain sine wave components are computed using equatio n [1]. Each is constructed such that at the midpoint of the pulse the sine wave passes through a maximum (or minimum if the coefficient is negative) at the same time. It should be noted that the "first" harmonic created using this formula is NOT the carrier frequency, f , of the modulated signal, but at F ± F AM While equ ation [1] is for rectangular waves only, similar equations can be constructed using Fourier coefficients for other waveforms, such as triangular, sawtooth, half sine, trapezoidal, and other repetitive geometric shapes. PRI Effects - If the PW remains constant but PRI increases, the number of sidelobes remains the same, but the number of spectral lines gets denser (move closer together ) and vice versa (compare Figure 7 and 8). The spacing between the spectral lines remains constant with constant PRI. Pulse Width (PW) Effects - If the PRI remains constant, but the PW increases, then the lobe width decreases and vice versa. If the PW approaches PRI, the spectrum will approach "one lobe", i.e., a single spectral line. The spacing of the lobes remains constant with constant PW. RF Measurements - If the receiver bandwidth is smaller than the PRF, the receiver will respond to one spectral line at a tim e. If the receiver bandwidth is wider than the PRF but narrower than the reciprocal of the PW, the receiver wil respond to one spectral envelope at a time. Jet Engine Modulation (JEM) Section 2-6 addresses the Doppler shift in a transmitted radar signal caused by a moving target. The amount of Doppler shift is a function of radar carrier frequency and the speed o the radar a nd target. Moving or rotating surfaces on the target will ha ve the same Doppler shift as the target, but will als impo se AM on the Doppler shifted return (see Figure 10) Reflections off rotating jet engine compressor blades, aircraf prop ellers, ram air turbine (RAT) propellers used to powe aircraft pods, helicopter rotor blades, and protruding surface of autom obile hubcaps will all provide a chopped reflection of the impinging signal. T he reflections are characterized by both positive and negative Doppler sidebands corresponding to the blades moving toward and away from the radar respectively. Therefore, forward/aft JEM doesn't vary with rada carrier frequency, but th e harmonics contained in the sidebands are a function of the PRF of the blade chopping action and its amp litude is target aspect dependent, i.e. blade angle int ake/exhaust internal reflection, and jet engine cowling al effect lateral return from the side. If the aspect angle is too far from head-on or tail-on and the engine cowling provides shielding for the jet engine, there may not be any JEM to detect. On t he other hand, JEM increases when you are orthogonal (at a right angle) to the axis of blade rotation. Consequently for a fully exposed blade as in a propeller driven aircraft or helicopter, JEM increases with angle off the boresight axis of the prop/rotor.