David83
Advanced Member level 1
Hello,
In OFDM systems, the N-by-1 data vector \[\mathbf{X}\] is first transformed to the time domain as \[\mathbf{x}=\mathbf{X}\mathbf{F}_N^H\], where \[\mathbf{F}_N^H\] is the N-by-N IDFT matrix. If we arrange the result in N-by-1 vector we can write \[\mathbf{x}=[\sum_{k=0}^{N-1}X_k\,\,\sum_{k=0}^{N-1}X_ke^{j\frac{2\pi}{N}k} .... \sum_{k=0}^{N-1}X_ke^{j\frac{2\pi}{N}k(N-1)}]^T\]. In the continous-time we have the passband signal as \[x(t)=\Re\left\{\sum_{k=0}^{N-1}X_ke^{j2 pi f_k t}\right\}\], where \[f_k=f_0+k/T\]. How we get from the discrete form to the continuous form?
In particular, if I have the signal in the discrete form as \[\mathbf{x}=[\sum_{l=0}^{L-1}X_{lM}....\sum_{l=0}^{L-1}X_{M-1+lM} .... \sum_{l=0}^{L-1}X_{lM}e^{j\frac{2\pi}{L}l(L-1)}....\sum_{l=0}^{L-1}X_{M-1+lM}e^{j\frac{2\pi}{L}l(L-1)]^T\], which corresponds to vector OFDM, how to write the continuous-time form?
Thanks
In OFDM systems, the N-by-1 data vector \[\mathbf{X}\] is first transformed to the time domain as \[\mathbf{x}=\mathbf{X}\mathbf{F}_N^H\], where \[\mathbf{F}_N^H\] is the N-by-N IDFT matrix. If we arrange the result in N-by-1 vector we can write \[\mathbf{x}=[\sum_{k=0}^{N-1}X_k\,\,\sum_{k=0}^{N-1}X_ke^{j\frac{2\pi}{N}k} .... \sum_{k=0}^{N-1}X_ke^{j\frac{2\pi}{N}k(N-1)}]^T\]. In the continous-time we have the passband signal as \[x(t)=\Re\left\{\sum_{k=0}^{N-1}X_ke^{j2 pi f_k t}\right\}\], where \[f_k=f_0+k/T\]. How we get from the discrete form to the continuous form?
In particular, if I have the signal in the discrete form as \[\mathbf{x}=[\sum_{l=0}^{L-1}X_{lM}....\sum_{l=0}^{L-1}X_{M-1+lM} .... \sum_{l=0}^{L-1}X_{lM}e^{j\frac{2\pi}{L}l(L-1)}....\sum_{l=0}^{L-1}X_{M-1+lM}e^{j\frac{2\pi}{L}l(L-1)]^T\], which corresponds to vector OFDM, how to write the continuous-time form?
Thanks