Gaussian pulses for 60GHz.doc

1. Direct pulsea. The ideal pulseThe ideal pulse can be derived from the Fourier transform of the frequency MASK.The ideal PSD is b. Gaussian series gj(t) is the Gaussian pulse (j=0) or its jth derivatives (j1) which can be described as各级导数 (1)where the amplitude A is used to normalize the pulse amplitude, and is pulse shaping factor. gg(f) is the power density spectrum of g(t), and it can be given by (2)where G(f) is the Fourier Transform (FT) of g(t). The FTs of Gaussian pulse series are shown as (3)Then we can get the PSD of the Gaussian pulse series with expression (2) and (3), (4)Fig 1. =36ps, k=15, i.e. 15th derivative of Gaussian pulseFig 2. =36ps, PSD of derivative of Gaussian pulseFig 3. =45ps, Gaussian derivative pulse series.Fig 4. =45ps, PSD of derivative of Gaussian pulse.Fig 3 and Fig 4 show that there is little difference in derivatives of Gaussian pulse in terms of shapes of their PSD and bandwidth.c. Hann window pulse,where fs is the modulation frequency that primarily impacts the bandwidth of the transmitted signal, fs=1/ Tp. d. Rectangulare. TriangularThe center frequency of Hann window, Rectangular and Triangular pulse is 0 GHz. These direct pulses can not meet the 60GHz spectrum requirements.f. Orthogonal modified Hermite pulseHermite polynomials can be modified to become orthogonal as (5)Where n=0,1,2,and -t1) which can be described as (8)where the amplitude A is used to normalize the pulse amplitude, and is pulse shaping factor.The PSD of the signal s(t)=p(t)cos(2fct), ss(f), follows the form 11, p204 (9)where pp(f) is the power density spectrum of p(t), and it can be given by (10)where P(f) is the Fourier Transform (FT) of p(t). The FTs of Gaussian pulse series are shown as (11)Then we can get the PSD of the modulated Gaussian pulse series with expression (3) and (5), (12)Finally, the PSD of modulated Gaussian pulse series can be given as (13)Figure 1 shows the modulated Gaussian pulse series with normalized amplitude when j=0, 1, 2 and their PSDs.(a) (b)(c) (d)Fig 7 a modulated Gaussian pulse (a), a modulated 1st derivative (b) and 2nd derivative (c) of the Gaussian pulse and their PSD (d), fc=60.5GHz, =1.1ns, Ap=1.0113.Fig 7 shows the time and frequency domain characteristics of a modulated Gaussian pulse and its derivatives. Figure (a), (b) and (c) describe pulse shapes of the modulated Gaussian pulse and its 1st and 2nd derivatives in time domain; (d) compares the PSD of the modulated Gaussian pulse and its derivatives with IEEE transmit spectral mask. It is clear that, with frequency shifting, the Gaussian pulse not only has no DC component, but also meets the transmit spectral mask better than its derivatives. Since there is a gap in the center frequency, the PSD of Gaussian derivatives can not fully utilize the specified spectrum resources. Although the PSD of Gaussian derivatives can meet the transmit spectral mask by changing , the relative -10dB bandwidth of Gaussian derivatives is much smaller than Gaussian pulse. Fig 8 modulated Gaussian pulses with different Larger corresponds to longer pulse duration. The pulse with long pulse duration has a narrow PSD and small bandwidth.B. Hann window pulse A pulse generated by modulating an RF carrier at the desired center frequency fc with a Hann window series can also meet the spectrum requirement, which can be described as (14)where fc is the center frequency, w(t) is Hann window function, which can be written as, (15)where fs is the modulation frequency that primarily impacts the bandwidth of the transmitted signal, fs=1/ Tp. The PSD of the Hann window pulse can also be written as (3), and the PSD of Hann window function hh(f) can be calculated as (16)where Sa(x)=sin(x)/x. (a) (b)Fig 9 a Hann window pulse (a) and its PSD (b), fc=60.5GHz, fs = 1/(2.2ns)Fig 9 describe a Hann window pulse and its PSD. The Hann pulse duration is 2.2 ns. This figure shows that the PSD of a Hann window pulse can also meet the transmit spectral mask.A hamming window pulse and a cosine window pulse have the similar pulse shape and PSD characteristics with a hann window pulse which can be described as follows respective- ly, (17) (18)C. Rectangular window pulseThe rectangular window pulse, which can be described as (13), can meet the PSD restrictions as well after up-converted to the center frequency. (19)where Tp is the pulse duration.The PSD of the rectangular window pulse can be written as (3), and the PSD of rectangular window function rr(f) can be calculated as (20)The rectangular window pulse and its PSD are shown in figure 3. With the same pulse time duration, the PSD of rectangular pulse does not fit the transmit spectral mask. But if the pulse duration increases, the PSD can also meet the spectrum regulations, but the relative bandwidth is small which means a waste of spectrum resources.Fig 10 A rectangular window pulse and its PSD, Tp=2.2 ns.D. Triangular window pulse A triangular window pulse can be written as (21)where fs=1/ Tp, is the modulation frequency that primarily impacts the bandwidth of the trans- mitted