使用Python和Numpy轻松实现根升余弦（RRC）滤波器的方法

Question

SciPy和Numpy好像支持很多种滤波器，但就是没有根升余弦滤波器。有没有什么简单的方法可以创建一个，而不是去计算传递函数？即使是个近似的也可以。

Answer 1

commpy 似乎还没有正式发布。不过我可以分享一些我的小知识。

beta = 0.20 # roll off factor

Tsample = 1.0 # sampling period, should at least twice the rate of the symbol

oversampling_rate = 8 # oversampling of the bit stream, this gives samples per symbol
# must be at least 2X the bit rate

Tsymbol = oversampling_rate * Tsample # pulse duration should be at least 2 * Ts
span = 50 # number of symbols to span, must be even
n = span*oversampling_rate # length of the filter = samples per symbol * symbol span

# t_step must be from -span/2 to +span/2 symbols.
# each symbol has 'sps' number of samples per second.
t_step = Tsample * np.linspace(-n/2,n/2,n+1) # n+1 to include 0 time

BW = (1 + beta) / Tsymbol
a = np.zeros_like(t_step)

for item in list(enumerate(t_step)):
    i,t = item 
    # t is n*Ts
    if (1-(2.0*beta*t/Tsymbol)**2) == 0:
        a[i] = np.pi/4 * np.sinc(t/Tsymbol)
        print 'i = %d' % i
    elif t == 0:
        a[i] = np.cos(beta * np.pi * t / Tsymbol)/ (1-(2.0*beta*t/Tsymbol)**2)
        print 't = 0 captured'
        print 'i = %d' % i 

    else:
        numerator = np.sinc( np.pi * t/Tsymbol )*np.cos( np.pi*beta*t/Tsymbol )
        denominator = (1.0 - (2.0*beta*t/Tsymbol)**2)
        a[i] =  numerator / denominator

#a = a/sum(a) # normalize total power

plot_filter = 0
if plot_filter == 1:

    w,h = signal.freqz(a)
    fig = plt.figure()
    plt.subplot(2,1,1)
    plt.title('Digital filter (raised cosine) frequency response')
    ax1 = fig.add_subplot(211)
    plt.plot(w/np.pi, 20*np.log10(abs(h)),'b')
    #plt.plot(w/np.pi, abs(h),'b')
    plt.ylabel('Amplitude (dB)', color = 'b')
    plt.xlabel(r'Normalized Frequency ($\pi$ rad/sample)')

    ax2 = ax1.twinx()
    angles = np.unwrap(np.angle(h))
    plt.plot(w/np.pi, angles, 'g')
    plt.ylabel('Angle (radians)', color = 'g')
    plt.grid()
    plt.axis('tight')
    plt.show()


    plt.subplot(2,1,2)
    plt.stem(a)
    plt.show()

Answer 2

如果能把根升余弦滤波器标准化到一个通用的包里，那就太好了。在这之前，我根据commpy做了一个实现。这个实现使用了numpy进行向量化处理，并且在归一化时没有考虑符号速率。

def raised_root_cosine(upsample, num_positive_lobes, alpha):
    """
    Root raised cosine (RRC) filter (FIR) impulse response.

    upsample: number of samples per symbol

    num_positive_lobes: number of positive overlaping symbols
    length of filter is 2 * num_positive_lobes + 1 samples

    alpha: roll-off factor
    """

    N = upsample * (num_positive_lobes * 2 + 1)
    t = (np.arange(N) - N / 2) / upsample

    # result vector
    h_rrc = np.zeros(t.size, dtype=np.float)

    # index for special cases
    sample_i = np.zeros(t.size, dtype=np.bool)

    # deal with special cases
    subi = t == 0
    sample_i = np.bitwise_or(sample_i, subi)
    h_rrc[subi] = 1.0 - alpha + (4 * alpha / np.pi)

    subi = np.abs(t) == 1 / (4 * alpha)
    sample_i = np.bitwise_or(sample_i, subi)
    h_rrc[subi] = (alpha / np.sqrt(2)) \
                * (((1 + 2 / np.pi) * (np.sin(np.pi / (4 * alpha))))
                + ((1 - 2 / np.pi) * (np.cos(np.pi / (4 * alpha)))))

    # base case
    sample_i = np.bitwise_not(sample_i)
    ti = t[sample_i]
    h_rrc[sample_i] = np.sin(np.pi * ti * (1 - alpha)) \
                    + 4 * alpha * ti * np.cos(np.pi * ti * (1 + alpha))
    h_rrc[sample_i] /= (np.pi * ti * (1 - (4 * alpha * ti) ** 2))

    return h_rrc

Answer 3

commpy这个包里有几个过滤器。早期版本中返回变量的顺序发生了变化（截至目前，这个包的最新版本是0.8.0）。要安装它，可以按照这里或这里的说明进行操作。

下面是一个关于1024个QAM16符号的例子：

import numpy as np
from commpy.modulation import QAMModem
from commpy.filters import rrcosfilter
N = 1024  # Number of symbols. Also, the filter length in samples.
os = 8    # Over-sampling factor
# Create modulation. QAM16 makes 4 bits/symbol
mod1 = QAMModem(16)
# Generate the bit stream for N symbols
sB = np.random.randint(0, 2, N*mod1.num_bits_symbol)
# Generate N complex-integer valued symbols
sQ = mod1.modulate(sB)
sQ_upsampled = np.zeros(os*(len(sQ)-1)+1,dtype = np.complex64) 
sQ_upsampled[::os] = sQ
# Create a filter with limited bandwidth. Parameters:
#      N: Filter length in samples
#    0.8: Roll off factor alpha
#      1: Symbol period in time-units
#     os: Sample rate in 1/time-units
sPSF = rrcosfilter(N, alpha=0.8, Ts=1, Fs=os)[1]
# Analog signal has N/2 leading and trailing near-zero samples
qW = np.convolve(sPSF, sQ_upsampled)

接下来解释一下这些参数。N是波特样本的数量。对于QAM来说，你需要的比特数是样本数的4倍。我让sPSF数组返回N个元素，这样我们就可以看到信号的前后样本。关于参数alpha的解释，可以查看维基百科的根升余弦滤波器页面。Ts是符号周期（以秒为单位），Fs是每个Ts的滤波样本数量。我喜欢假装Ts=1，这样事情就简单了（单位符号率）。然后Fs就是每个波特点的复波形样本数量。

如果你从rrcosfilter中获取返回元素0来获取样本时间索引，你需要在Ts和Fs中插入正确的符号周期和滤波样本率，这样索引值才能正确缩放。