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Equalization, Convolution, and Cyclic Prefix Addition

This example introduces frequency domain equalization and shows how to convert circular convolution to linear convolution. When considering a linear channel model, the received signal is the convolution of the transmitted signal with the channel impulse response. In the frequency domain, the received signal Y(f) is the linear convolution of the transmitted signal U(f) with the channel impulse response H(f):

Y(f)=H(f)U(f)

OFDM receivers use frequency domain equalization to recover the original transmitted signal, so that:

U(f)=Y(f)H(f)

FFT processing yields the circular convolution of u with h. For the circular convolution of u and h to be equivalent to the linear convolution, u and h must be padded with zeros to a length of at least (length(u) + length(h) - 1) before you take the discreet Fourier transform (DFT). After you invert the product of the DFTs, retain only the first N + L - 1 elements. For an example that demonstrates this process, see the Linear and Circular Convolution topic.

Define a short input signal, u1, and channel impulse response, h. The input signal must be longer than the channel impulse response. Display a stem plot of the signals.

u1 = 1:8; 
h = [0.4 1 0.4];

figure
subplot(2,1,1)
stem(u1);
axis([0 10 0 10])
title("Input signal")
subplot(2,1,2)
stem(h);
axis([0 10 0 2])
title("Channel impulse response")

Figure contains 2 axes objects. Axes object 1 with title Input signal contains an object of type stem. Axes object 2 with title Channel impulse response contains an object of type stem.

Compare the circular and linear convolution of u1 with h. Perform linear and circular convolution by using the conv and cconv functions, respectively. The smearing effects due to the nonideal channel cause the linear and circular convolution to yield different results at some points. A cyclic prefix (CP) enables effective use of OFDM in a nonideal channel with unknown propagation delay.

N = length(u1);
yl1 = conv(u1,h);
yc1 = cconv(u1,h,N);
figure;
stem(yl1,"x")
hold on;
stem(yc1,"o")
title(["Convolution Results - N=",int2str(N)])
legend ("Linear","Circular","Location","northwest")

Figure contains an axes object. The axes object with title Convolution Results - N= 8 contains 2 objects of type stem. These objects represent Linear, Circular.

Add Cyclic Prefix (CP)

For OFDM processing, the necessary padding for the circular convolution is provided by adding a CP rather than zero-padding the signals. Adding a CP that repeats the end samples of the symbol enables:

  • Modeling of the linear convolution of a frequency-selective multipath channel as circular convolution

  • Use of an FFT to compute the convolution

  • Simple frequency domain processing for channel estimation, equalization, and synchronization

  • Repeated samples to be used in forward error correction schemes

L = length(h);      % Length of channel
N = length(u1);     % Length of input signal
ucp = u1(N-L+1:N);  % Use last samples of input signal as the CP
u2 = [ucp u1];      % Prepend the CP to the input signal
yl2 = conv(u2,h);   % Convolution of input+CP and channel
yl2 = yl2(L+1:end); % Remove CP to compare signals

figure;
stem(yc1,"x")
hold on;
stem(yl2,"o")
title("Convolution Results with Cyclic Prefix")
legend ("Linear","Circular","Location","northwest")

Figure contains an axes object. The axes object with title Convolution Results with Cyclic Prefix contains 2 objects of type stem. These objects represent Linear, Circular.

Compare the linear and circular convolution sequences.

if max(yc1 - yl2(1:N)) < 1e-8
    disp("Linear and circular convolution sequences match.")
else
    disp("Received symbols do not match transmitted symbols.")
end
Linear and circular convolution sequences match.

See Also

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