矩形脉冲序列FFT频谱仿真异常问题排查求助

我现在需要模拟一组矩形脉冲序列，于是写了一段Python代码，但最终的频谱结果不太理想。从生成的图里能看到，脉冲序列的波形是符合预期的，但频谱部分明显有问题——我试过提高采样率也没能解决这个问题。想请教一下，我的代码里哪里出错了，导致FFT频谱生成异常？

#!/bin/python

import numpy as np
from numpy.fft import fft
import matplotlib.pyplot as plt

A = np.random.randint(1, 6) # random amplitude (from 1 to 5)
tilda = 0.5 # pulse duration
SR = 1000 # sampling rate
t = np.linspace(0, 3, SR) # time range from 0 to 2 seconds

n_pulses = 3 # number of pulses
t_i = 0.5 # interval between pulses
pulse1_start = 0.25 # the beginning of the first pulse

rect_pulses = np.zeros_like(t)
for i in range(n_pulses):
    pulse_start = pulse1_start + i * (tilda + t_i)
    pulse_end = pulse_start + tilda
    rect_pulses = np.where((t >= pulse_start) & (t <= pulse_end), A, rect_pulses)

FFT = fft(rect_pulses) # magnitude of FFT
FFT_n = FFT / len(FFT)  # normalization
f_ax = np.linspace(0, (SR), len(FFT)) # frequency axis 
half_fft = len(FFT) // 2 # half the length of the FFT
f_half = f_ax[:half_fft] # frequencies up to f_ax / 2
FFT_half = np.abs(FFT_n[:half_fft]) # FFT module up to f_ax/2
FFT_half *= 2 # doubling the magnitude 

plt.figure(figsize=(10, 10))
plt.subplot(2, 1, 1)
plt.plot(t, rect_pulses, linewidth=1.7)
plt.title('Rect. Pulse', fontsize=21)
plt.xlabel('t', fontsize=21, fontweight='bold')
plt.ylabel('x(t)', fontsize=21, fontweight='bold')
plt.grid()
plt.subplot(2, 1, 2)
plt.plot(f_half, FFT_half, 'r-', linewidth=1.7)
plt.title('FFT of Rect. Pulse', fontsize=21)
plt.xlabel('$f$', fontsize=21, fontweight='bold')
plt.ylabel('S($f$)', fontsize=21, fontweight='bold')
plt.xlim([0,40])
plt.grid()

plt.tight_layout()
plt.show()

备注：内容来源于stack exchange，提问作者Varga